As a general rule, ingredients are measured in weight rather than volume because this allows for better precision. On that topic, a digital scale is a must to make sure you are following the correct proportions.
Use flour with a high protein content, 12-13g per 100g. Protein is important to develop the gluten which keeps the dough elastic and stretchable.
In Finland, where I live, I use "erikoisvehnäjauho", which is the equivalent of all-purpose flour. You can find more than you could ever possibly want to learn about Finnish flour here.
Using very little yeast lenghtens the proofing time and gives time for the dough to develop flavour. Store-bought fresh or dry yeast is fine, natural yeast alone doesn't have enough strength to make the dough raise, but can be added for extra flavour.
Whatever recipe you use, using a tiny amount of yeast (and consequently a long proofing time) is one of the most important improvements you can make for the flavor and consistency of your dough.
I prefer whole peeled tomatoes to crushed ones because the taste is slightly better. Also, you may find canned San Marzano peeled tomatoes, and of course if you have the chance you should use fresh ones. The thing to remember about the sauce is that it shouldn't be too watery, otherwise the pizza will get soggy. That's why it is left to drain in a colander for a while.
Some recipes call for cooking the sauce first, but this is redundant because the tomato will cook in the oven in the pizza, and in fact may result in a burnt sauce.
Put all the water in a large container, dissolve the yeast in it and start pouring in the flour while mixing with a large spatula (or spatula attachment if using a kneading machine). When half of the flour is used, add the salt to the remaining flour and continue incorporating. Let rest for 15 minutes.
If using a kneading machine, switch to the dough hook attachment and knead at a low speed for 15 minutes. Otherwise transfer to the work surface and knead by hand. Knead for 15 minutes and let rest for another 15 minutes.
A wet dough (68-70% hydration) is difficult to work by hand, so a kneading machine is recommended. If you are going to knead by hand, dust the dough with a little flour often, but be careful not to use too much.
Knead the dough ball using the "stretch and fold" technique (see video below): stretch it, rotate 90 degrees, fold it in half, stretch it again and repeat. This makes the dough much more firm and elastic. Shape into a ball and store in a large plastic container previously coated with oil (drizzle a few drops and spread with a piece of kitchen paper). Store in the fridge and let rest overnight, or out of the fridge for 4-6 if you are short on time.
I found one night (or 24h) to be the proofing time that gives the best results without being too long - planning your pizza a day ahead seems like a reasonable demand. You can also let the dough rest for two nights if you want, but more than that will not produce any better result.
Divide into 4 chunks and shape into balls, let rest for 1.5 hours. Put the pizza stone in the oven and turn the oven on at max temperature. Let it heat for at least 30 minutes. Meanwhile, roughly blend the canned tomatoes with their juice and leave them into a colander to drain. Break the mozzarella with your hands and put in another sieve to drain.
Flatten and stretch one of the balls in to a pizza disc (see how in the video below), transfer to the pizza peel, spread one spoonful of tomato sauce on it and toss on top of the hot stone.
A pizza peel and a pizza stone are fundamental tools and will improve any pizza recipe tremendously. Your pizza needs to land on a very hot surface so that the "heat shock" will make it bubble and raise instantly. The cooking time needs to be fast so that the inside of the crust stays moist and airy, while the outside becomes crispy.
After 5-7 minutes the pizza should have risen but not be fully cooked yet, take it out, add the mozzarella, basil leafs, a drizzle of olive oil and return to the oven for another 5 minutes or less.
The cooking time depends a lot on the oven, you'll have to try a few times to find the one that works for you.
This recipe would not exist if it wasn't for Jeff Varasano and his Famous New York Pizza Recipe. I also owe a lot to Serious Eats' J.Kenji Lopez and his Pizza Lab. Thanks guys, you are an inspiration!