You've got this all mixed up! Unless you're deglazing to do a pan sauce, the steak should be the LAST thing cooked. It's the centerpiece of the dish and the most expensive part, and shouldn't have a chance to get cold while you cook the other parts. Saving cleaning on a single saucepan is not worth eating cold steak.
You MUST use a stainless or cast-iron for this. As others touch on, teflon is not safe for searing, and fast temperature changes will wreck it.
Here's how to do things:
- Julienne onions to saute, prepare and measure stock, trim fat from steak, cut or butterfly it if needed (this is your mise en place).
- Start preheating saute pan for onions
- Start butter melting in saucepan for roux
- Prepare roux and saute onions at same time, then add stock to roux, whisk, and season
- Transfer onions to bowl, cover to keep hot, and start to preheat pan for searing steak
- Cook steak while finishing sauce as needed, then rest it appropriately
- Plate steak, garnish, sauce, and serve
Alternately, here's another approach:
- Mise en place (prep ingredients, optionally pre-measure them)
- Prepare sauce in advance, set aside or keep hot. Or, even do it the day before and reheat it. It's a roux-based sauce; it won't go bad in the fridge, and will reheat beautifully.
- Saute onions. Or, caramelize onions beforehand, and just reheat them briefly and add.
- Cook steak
- Plate, sauce, serve
And, finally the deglazed pan sauce way
- Mise en place
- Sautee onions. Set aside in bowl (or on a serving plate)
- Cook steak
- Throw heavy cream or red wine in to pan, and deglaze by scraping vigorously with a utensil and stirring as it boils vigorously
- Quickly season this, and throw in the onions, a few pinches of parsley, pepper, salt, garlic, and paprika, plus optionally thyme
- Congrats, you now have a pan sauce with onions to throw on your steak
In a professional kitchen, you'd have your onions pre-sauteed or sauteeing at the same time as the steak, and your sauce will be prepared in bulk, beforehand. You'd only have to cook the steak and briefly heat or season the other two parts.
If you're boiling something rapidly, and it's not in a terribly deep, narrow pot, then essentially all of the heat output of the burner is going into turning water into steam. The latent heat of vaporization of water is 2260 kJ/kg, so if you want to reduce something by a volume V, and your stove has power P, the time required is:
t = V * (1 g/mL) * (2260 J/g) / P
If V happens to be in mL, and P is in W (J/s):
t (s) = V / P * 2260
This would be modified slightly if you're using a really tall, skinny pot, since the convection within the pot, from the bottom to the top of the liquid, would be less efficient, with more heat transferred to the sides of the pot and out into the air, but I doubt you're actually going to try to reduce something like that. The P here is the effective power; for example, a gas burner wastes a lot of heat out the sides, so the advertised power will be higher. See TFD's answer for approximate efficiencies.
If you don't know the power of your stove, in all honesty, the easiest way to measure it would probably be to just see how long it takes to boil away a given volume of water, and work backwards. To get an accurate result, you should not boil a pot dry - once the water is a thin enough layer, the heat transfer might start working differently, with the pot itself heating up more, and water splattering. So you could, for example, put in a liter of water, boil away at the stove setting you intend to measure until it's substantially reduced in volume, record the time, then pour it out to measure how much you boiled away. At this point, knowing the power output might be overkill, though; you can really just measure the time per volume reduction, and use that, unless you care about the power for other reasons.
Trying to deduce the power of the stove from, say, the temperature of an empty pot or of the burner without a pot on it (assuming it's electric) would be difficult; you'd have to deal with the heat transfer between metal and air, and the convection in the air.
Dependence on ingredients shouldn't be significant - you're still just boiling water, unless there's a substantial amount of alcohol, in which case the latent heat of vaporization will be different. Pure alcohol has a latent heat of vaporization of 841 kJ/kg; I haven't found a good table for mixtures.
For solutions, as I noted in the comments, the latent heat of vaporization should be that of water, plus/minus the heat of solution of the solutes (I forget which direction that's measured in). The most common solutes are probably salt and sugar, which have heats of solution of 70 and 16 J/g, respectively. (I found this table, and converted.) The next most common thing I could think of that might be present in substantial concentrations is citric acid; this paper reports a heat of solution of -57 J/g. In all these cases it's small compared to the latent heat of vaporization of water, so pretending the liquid is water should be a good approximation. It's possible that things change if you're reducing really far: heat of solution does depend on concentration. That is, things are different thermodynamically (statistical mechanically?) in a nearly-saturated sugar syrup than in slightly sweet water.
Best Answer
The biggest reason your sauce didn't thicken is that you didn't have much of anything at all in the pan that will gelatinize and help trap the water molecules present in the sauce. Starches (flour, cornstarch) will provide some of this, as will a liquid like stock that contains some dissolved collagens. But wine and water by themselves will have very little thickening power. At minimum, you'll want to add a teaspoon or two of cornstarch to your wine/water mix and shake in a covered vessel - this is called a slurry and will thicken fairly quickly, but you need to be careful not to overcook it.
For a more traditional (and in my opinion, much tastier) pan sauce, try the following steps:
First, make sure that you pour off any excess oil - you want no more than about a tablespoon or so left in the pan, less if you're adding butter.
Second, heat the pan by itself for a few moments, which will help dissolve the fond (little flavorful brown bits left in the pan) during the next steps. If you want to add your onions, this would be the time to do so.
At this point, if you want a really thick sauce, add your butter, and sprinkle in a very small amount (no more than the amount of butter you've added) of regular flour. Whisk this well for about a minute until it seems to thin slightly. What you're doing here is creating a quick roux that will help thicken the final mixture. You can skip this step if you like.
Next, carefully add your wine and wait a moment to help burn off some of the alcohol flavor. Whisk or stir with a fork for few seconds to help dissolve the flavorful fond. Then, add stock - this is very advisable especially if you didn't create a roux. Stock both has more flavor than water and (especially with meat stocks) contains collagen, producing a more velvety-tasting final sauce. Store-bought stocks are fine, and are a great pantry item to keep around; you don't have to make your own.
At this point, stir in your other flavorings (herbs, etc.) and reduce the sauce over medium-low heat until you're happy with the texture. A good way of checking is to dip in a spoon, flip it over, and run your finger across the back. If the sauce doesn't fill in the track left by your finger, it's good to go.