How To Properly Make a Pizza (Screen Cooking, Not Pan)
The first step is to make sure your dough is properly proofed. Improperly proofed dough makes this job much more difficult. If it's under-proofed, it's going to have a wait to shrink up on you. Over-proofed dough is just FULL of air, and as a result you're going to have a lot of bubbles if you're not extremely careful. Properly proofed dough is almost spongy and at least 25-50% larger than it started before proofing.
Step two is to prepare your work surface. You're going to want a flat, clear space at least a couple square feet. It's important to have enough room to work, because you're going to be cocking your elbows out a bit soon. You're going to need to put something down to ensure that the dough does not stick to the surface. Cornmeal, flour, really anything of that nature will do. An even, moderate distribution is really the key here.
Now you've got yourself set up with a nice work area, it's time to zoom in a bit. When you place a dough ball to proof, there's a section that's not exposed to the air. As a result, it does not develop a "skin" like the dough that IS exposed to the air. That's what you want for your edge crust, because it's going to bake into a much more soft, more pleasant bread. The best way to do this is to place your dough ball in the center of your prepared work area, with the previously exposed side up.
Take your hands and interlace your index fingers to form a V shape with your hands, middle fingers touching. The thumbs should be on top. You're going to want to arch your fingers slowly to ensure that you'll be pressing with the actual flat surface of the finger, rather than the tip. If you use the tip you're much more likely to create a sauce "trough" which creates a somewhat funky shaped crust. Cock your elbows out to about a 45 degree angle, and move your finger tips about a quarter to half an inch beyond the edge of the dough ball.
This next part is possibly the most difficult part to learn, because your body does not yet know how exactly to move that way. You want to pull back while also pushing down, but you do not want to pull back with your hands. Your entire arm should supply the motion and force. Bring your hands towards yourself while steadily increasing pressure until the bottom edge of the dough ball rolls up towards you. You should create almost a quarter pipe ramp with the motion.
Rotate the dough ball one fifth to one quarter (depending on the size of your hands and the spread you have with the V shape) and repeat until you've gone all the way around the dough ball. This creates the locked edge and defined crust. What you're going to want to do next is to reinvent that line by pressing down with the side of your hand all around the outside edge where you've just made the indentation.
Once this is done a couple times around, flip the dough ball (now somewhat of a dough disc) over. Since dough is rather pliable, you can see the indentation you made on the other side clearly. At this point you're going to have to stretch the dough. Your hands should never be more than 2 inches apart during a stretch. Starting with the tips of the fingers right up to the defined crust line and index fingers touching, you're going to want to pull with your non-dominant hand 1 to 2 inches to the side, keeping your dominant hand right up to that crust line.
At the very end of the pull you can let the dough ball slide a bit under your dominant hand, turning it in the same motion. Then simply repeat the stretching motion in the same short, somewhat quick tempo. You do have to find the right balance between sharp motion and where the dough will actually tear, of course. But without the sharp stretch, the air will not be properly discharged from around the crust area and you're going to see bubbles during the cooking process. Most dough consistencies that I've worked with only need to be worked around 3 to 5 times with the stretching motion. It does take some practice, so do not get discouraged.
Now either you have the dough to the proper size for the pizza you want to make, or it's much too small. This is where there are two personal choices to be made. One, some people like to stretch the dough on the table the whole time while others actually pick it up and toss it. I have found the more effective way is to actually hand toss the dough, so if you want to attempt this you're going to want to leave the dough around half to a third the size you want your finished product to be. Actually tossing dough requires a LOT of practice to do well, but the pizzas made using this entire method are consistently almost perfect. The second personal choice is about docking the dough. Some people support it, others are very against it. I'm personally for it, as it can help rid the dough of any remaining small air pockets you may have missed.
To toss the dough, you're going to pick it up by the edge, inside the crust line. Place that line on the edge of your dominant hand, because that first throw is the part where you're really setting up the rhythm and you want to have the most control. Now, and again I stress this requires a lot of practice (some people like to use cloths to get the motion down), you want to pull the dough up over your left hand, so that the catch catches on the tips of your non- dominant hand, just at the crust line. Pulling farther stretches the dough. The idea is to get used to this motion back and forth so that you can actually throw the dough between the hands.
Because you're coming across one side to the other, it's a simple step to turn the dough slightly as you're tossing. In this way, you get an even stretch across the dough surface. If not, you're going to wind up with a very lop-sided pizza. When the dough is the proper size, simply fit it to the screen with the side that was not exposed to the air during the proofing process up.