Dissolve the yeast in the warm water and set aside (a Pyrex 2-cup measure makes for easy pouring; be sure the cup isn't cold).
Meanwhile, put the flour and salt in a food processor fitted with the steel blade; process briefly to mix.
With the machine running, add the water-yeast mixture in a steady stream.
Turn the processor off and add the oil. Pulse a few times to mix in the oil.
Scrape the soft dough out of the processor and onto a lightly floured surface.
With lightly floured hands, quickly knead the dough into a mass, incorporating any bits of flour or dough from the processor bowl that weren't mixed inches
Cut the dough into four equal pieces with a knife or a dough scraper. Roll each piece into a tight, smooth ball, kneading to push the air out.
What you do next depends on whether you want to make pizza right way or at a later date.
If you want to bake the pizzas as soon as possible, put the dough balls on a lightly floured surface, cover them with a clean dishtowel, and let them rise until they almost double in size, about 45 minutes.
Meanwhile, turn your oven on, with the baking stone in it, to let the stone fully heat.
If you want to bake the pizzas tomorrow, line a baking sheet with a floured dishtowel, put the dough balls on it, and cover them with plastic wrap, giving them room to expand (they'll almost double in size), and let them rise in the refrigerator overnight.
To use dough that has been refrigerated overnight, simply pull it out of the refrigerator about 15 minutes before shaping the dough into a pizza.
To freeze the dough balls, dust each one generously with flour as soon as you've made it, and put each one in a separate zip-top bag. Freeze for up to a month.
It's best to transfer frozen dough from the freezer to the refrigerator the night before (or 10 to 12 hours before) you want to use it. But I've found that dough balls pulled straight from the freezer and left to warm up on the counter will be completely defrosted in about 1-1/2 hours. The dough is practically indestructible.
Put the proofed or thawed ball of dough on a lightly floured wooden board. Sprinkle a little more flour on top of the ball. Using your fingertips, press the ball down into a flat cake about 1/2 inch thick.
Lift the dough and lay it over the back of the fist of one hand. Put your other fist under the dough, right next to your first fist. Now gently stretch the dough by moving your fists away from each other (see Video). Each time you do this stretch, rotate the dough. Continue stretching and rotating until the dough is thin, about 1/4 inch, and measures about 9 inches across. Unless your dough is still cold from the freezer, it will be so soft that its own weight will stretch it out. Alternatively, use a rolling pin to roll out the dough thinly on a floured board. If you like a very thin pizza, roll the dough out to a 10-inch round. Be careful not to make it too thin, and remember that the thinner the pizza, the less topping it can handle.
Rub a bit of flour onto a wooden pizza peel (or the back of a baking sheet). Gently lift the stretched dough onto the floured peel. Top the pizza, scattering the ingredients around to within 1/2 inch of the border.
Topping your pizza.
For some people, pizza isn't pizza without the scarlet of tomatoes peeking through the cheese, but there are many delicious savory combinations that show off fresh seasonal produce. It's better to use winter vegetables like greens or even canned tomatoes when fresh tomatoes are out of season.
To get you started, here are two of my favorite ways to top a pizza -- plus lots of suggestions for combinations to inspire your own designs.
To make the Angeli Caffé's favorite, Pizza al Caprino -- Over the shaped pizza, scatter 10 to 15 cloves roasted or slow-cooked garlic, 5 to 6 oil-packed sun-dried tomatoes (drained and sliced), 3 ounces crumbled goat cheese, a few capers, and a pinch of oregano. Drizzle with extra-virgin olive oil.
To make a simple flatbread -- Scatter sliced garlic (3 to 4 cloves), minced fresh rosemary (from 1 small sprig), and coarse salt over the dough. Make several 1/2-inch slashes to keep the dough from puffing up. Drizzle with lots of extra-virgin olive oil before baking, and garnish with Parmesan. Serve this delicious "Pizza Aglio e Olio" with a salad or cheese.
To design your own pizza -- Use any of these topping combinations to inspire your own creation. A generous drizzle of olive oil is a great addition to just about any pizza.
Put a pizza stone or unglazed terra-cotta tiles on the lowest rack of the oven and heat the oven to 500°F Ideally, let the stone heat in the oven for an hour.
Shake the peel (or baking sheet) gently back and forth to make sure the pizza isn't stuck. If it seems stuck, lift the edges up with a spatula and toss a bit of flour under the dough. Quickly slide the pizza onto the hot baking stone. Bake until the edges are golden, about 8 minute Using a peel, a wide spatula, or tongs, remove the pizza from the oven.