Your filling is basically a beef stew/casserole, so any cuts that suite long slow cooking are fine. They get cooked again while the pie is baked, so premium cuts will just be wasted
The lower grade and 'off cut' parts are what are most commonly used. Like rump, brisket, chuck, shank, neck etc
Using a bench top slow cooker makes cooking this all very easy. The beef needs to be fully cooked before making the pie. An eight hour slow cook is fine
You want the pieces to be a variety of sizes to fill out the pie shape, but make sure you have some at least the height of the pie filling. The larger pieces tend to be better, but that could be a matter of taste and culture
You need to get rid of plenty of fat before and after cooking. A mouth full of fat in a pie is not nice. There is already plenty of fat in the gravy and pastry, so trim off all you can
A good meat pie should have a solid layer of meat connected with gravy, not swimming. It should also not be so full as you can't eat it with your hands (for a single serve pie) without loosing half the contents
Puff/flakey pastry works best and holds everything together
Avoid overly strong flavoured gravies, you don't want to hide the beef and pastry taste
Before you attach the lid, place a slice of tasty cheese, or an extra generous serving of cracked pepper on top of the filling
A respectable pie looks like this - thanks to Ponsonby Pies (not made in Ponsonby anymore)
"Stew beef" is slightly cheaper than buying a whole roast and cutting it up, because the stew beef is made up of bits and pieces that were left over after the prettier roasts had been carved.
If it's not to your taste, spring for a whole roast and cut it up yourself.
WARNING: Fat content in meat that is supposed to be cooked for a long time is a good thing. Keeps the meat from drying out. It's more efficient to simply cook the stew in advance, and then skim the rendered fat off the top, before you reheat it.
Best Answer
The most obvious thing is it keeps your oven free; a range has only one oven, but four or so burners. Its also often easier to check on it, add ingredients, etc. stove-stop. Stove-top also lets you quickly turn up or down the heat. Stirring is easy stove-top, more annoying in the oven.
For what you're doing, the biggest difference is going to be where the heat comes from. Stovetop, of course, comes almost entirely from the bottom of the pot, the lid for example can remain fairly cool, meaning moisture evaporating can condense on it, and drip back in. In the oven, the heat surrounds the pot, so the lid will be just as hot—or hotter, even—than the base of the pot.
To take a recipe designed for stovetop, you'd perform all the browning steps stovetop (of course). Maybe add a little more liquid. Then bring it to a simmer stovetop, and finally transfer to the oven.
Putting a piece of aluminum foil over the pot (under the lid) can help keep moisture in.