Mix sugar, yeast, oil, and 2 cups cold tap water in the bowl of a stand mixer fitted with a dough hook. Let sit 10 minutes for yeast to dissolve and reactivate. To mix by hand, combine the ingredients in a large bowl.
In a separate bowl, combine flour and salt. "The biggest mistake people make when mixing dough is to put the salt with the yeast at the beginning. It kills the yeast, and the dough will never proof." On low speed, gradually spoon flour mixture into yeast solution and mix until dough is elastic, fairly smooth, and you can pull it cleanly off the dough hook, 15 minutes (you may need to add up to 1 cup more flour if it's too sticky). To mix by hand, stir flour mixture into yeast until well combined. Knead in the bowl until a ball of dough forms, sprinkling in more flour as needed to keep dough from sticking to your hands. Turn out onto a floured work surface and knead vigorously until dough is elastic and fairly smooth, about 15 minutes. Tuck edges of dough under to form a tight ball.
Transfer dough ball to a clean bowl and cover with a wet but not dripping kitchen towel, shaping it around dough. Let rise 1 hour. "This resting period makes the flavor more complex."
PORTION OUT DOUGH
Scrape dough out onto a work surface and cut into 6 equal pieces. Working with 1 piece at a time and keeping others covered with a wet towel, tuck ends of piece into center to form a ball. Set ball, ends side down, on surface. Rotate gently but firmly against surface to make it tighter and smoother. Set ball on a well-floured rimmed baking pan. Drape with a wet towel. "The dough must stay humid. If a crust forms, it will be extremely difficult to stretch." Roll rest of dough the same way, spacing balls a few inches apart on baking pan.
Let balls rise at room temperature, making sure towel stays damp, until they have almost doubled in size, at least 3 hours. They will be ready to use for up to 5 more hours, depending on room temperature. (Warmth makes dough rise faster.) If dough overproofs--it will be bubbly, very soft, and hard to stretch--roll back into a ball with a little extra flour to feed the yeast, and let rise again.
STRETCH AND TOP PIZZA
First, measure your toppings and line them up, ready to go. Stretched crusts need to be topped quickly or they'll dry out.
Place pizza stone on rack in oven and preheat to 550° (or as high as your oven will go). Once oven reaches 550°, let stone heat another 15 minutes.
Working with 1 dough ball at a time, dip both sides into flour. Set on a floured work surface. With pads of fingers, pat dough out from center to make a circle 8 in. wide, leaving a 1-in.-thick "frame," or rim, around the edge. "This is the most important step in the pizza: to push air to the edges to make the cornicione, the frame."
Place a hand inside the circle with your thumb in the center. With the other hand, grasp opposite side of circle (within rim). Stretch circle gently outward, then flip it over other hand. Leaving dough over hand, rotate it a bit and unfold onto work surface. If dough tears, pinch it firmly together. Repeat several times until you have a smooth, round crust 9 to 10 in. wide.
Drape circle over backs of both hands and gently stretch outward, rotating a couple of times. "Twirling the dough in the air? That's not Naples. This is Naples." Lay dough on work surface. With your fingertips, push into an even circle about 10 in. wide.
Working quickly, layer toppings onto dough. Lightly flour a metal pizza peel and set next to pizza. Quickly drag pizza onto peel, trying not to crush the rim. "If you wait more than a minute or two, especially if it has tomato, you cannot move the pizza." With your fingertips, restore the pizza's shape.
Jiggle pizza peel to make sure the pizza is sliding freely. If it isn't, slide it back onto floured work surface, sprinkle peel with a bit more flour, and drag pizza back onto peel. "We don't make the pizza right on the peel. You need too much flour to keep it from sticking, and the flour goes into the oven, burns, and gets bitter."
Jiggle pizza onto pizza stone as you pull peel out from underneath. Bake until crust is puffed and charred in spots, 7 to 10 minutes. Transfer to a wooden board, finish with remaining toppings, if any, and serve immediately.
Make ahead: Chilled, up to 1 day: Make dough balls through step 4 and chill, baking pan covered with plastic wrap. Before stretching, replace plastic with a wet towel and let rise at room temperature until almost doubled in size. Frozen, up to 1 month: Freeze dough balls until hard and then wrap each well. Before stretching, let balls thaw on a baking pan in refrigerator overnight, then replace plastic with a wet towel and let rise at room temperature until almost doubled in size.