@Adam A is close -- it's not an issue of surface area on potency, it's an issue of damage to the garlic. The 'strong' taste of garlic comes from a reaction as chemicals are released so they can mix (alliin and alliinase)
When you cook the garlic whole (as you would for roasted garlic), you will never get this reaction, as you'll break down the chemicals. Also, the chemicals break down over long cooking, so even if you add a head of crushed garlic at the beginning of a batch of slow-cooked tomato sauce (4+ hrs), it's not going to have as strong a garlic flavor as adding a clove or two at the end.
One other way that the garlic prep can affect the taste is when you're sauteing, stir frying or other cooking over high heat -- larger bits can be cooked longer before they burn ... and burned garlic is bitter, acrid and will ruin any dish. (if you burn garlic, stop immediately, trash everything, clean the pan, and start again -- there is no way to save it that I know)
I can't give a specific temperature, as there's issues:
- The sharpness is a chemical reaction which mix when the garlic cells are damaged.
- Cooking the garlic before damaging the cells will convert the chemicals before they've had a chance to react.
- Cooking the garlic after the chemical reaction will also remove the sharpness.
- I've never tried taking the temperature of garlic as it's cooking
Now, if you're cooking the garlic cloves ahead of time (I roast a couple of bulbs at a time, then squeeze out the cloves into small jar and keep it in the fridge). It'll take 30min to an hour, depending on what temperature you're roasting at (350-400F; I'll throw it in with something else ... not worth heating the oven just for garlic)
If cooking after you've cut it, it'll only take a quick saute as you have more surface area (maybe 1 to 2 min, depending on pan temp) -- but be careful, as overcooking garlic will burn it, which is just nasty. And don't do it in a dry pan, unless you're planning on ruining your dinner. When it's golden, you're done ... don't let it get to brown, as brown is that stage right before black and time to clean the pan before starting again.
Also, it seems counter-intuitive, but finely minced garlic is more potent than coarsely chopped garlic, as you're doing more damage to the garlic. It won't help in this particular case, but in many dishes that just want the garlic flavor without being overpowering will use more garlic, but slice it rather than mincing, saute it in olive oil, then remove the garlic pieces, and use the garlic-infused oil for the dish.
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So, anyway, not not exactly answer the questions ... I'd go with low and slow cooking ... specific temperatures are going to lead to fast cooking garlic, which gets you burnt garlic and ruined dishes.
In dishes where you can, infuse the oil, then remove the garlic. And if you really have to, there's always garlic powder.
Best Answer
You will find things much easier if you cut the garlic in half at it's widest point. If you cut the tops off like in the link provided then you are restricting the cloves from coming out of their wrappers. I think the point with cutting the tops off is to be able separate the cloves and use them as individual applicators, kind of like ketchup packets in a way. Cutting the whole thing in half means that the garlic should pop out with a gentle squeeze.
I have used the flat of a chef's knife to squeeze the garlic out rather than using my hands and it works pretty well. The garlic needs to be very done for this method or using your fingers work though.