In a large bowl, combine the flour, yeast, salt, and sugar. Stir with a wooden spoon or dough whisk until everything is mixed together.
Pour in the warm water and stir with the wooden spoon or dough whisk until a shaggy dough forms and there is only a little bit of dry flour at the bottom of the bowl.
If you're wearing any rings on your fingers, take them off. Knead the dough in the bowl or on a floured work surface with both hands until no more dry flour remains in the bottom of the bowl and the dough comes together into a ball. Start by poking the dough with your fingertips and squishing it through your fists, and then, when the dry flour is incorporated, keep folding and pressing it down with the heel of your hand until the dough doesn't stick to your hands and stays together in a smooth-ish ball. (This whole process should only take a few minutes.)
Grab a large bowl (preferably one that has a lid) and drizzle in some olive oil to coat the inside. (Or, you could clean the bowl you made the dough in and use that.) Place the dough ball in the greased bowl, cover tightly with the lid or plastic wrap, and put it in a warm place to rise for at least 45 minutes or up to 2 hours, until it has not quite doubled in volume and has little bubbles on the bottom.
Let your oven get super-hot while the dough rises. If you're baking the pizzas on a baking sheet or pizza pan, preheat the oven to 500°F. If you're using a baking stone or steel, place it in the bottom third of your oven and preheat to the hottest-possible temperature (my oven goes up to 550°F).
Dust a clean work surface lightly with flour. Divide the dough into two equal pieces, and form each piece into a ball by stretching the edges underneath and grabbing the bottom of the ball with your fist. Place the dough balls on your floured work surface and cover with an inverted bowl or a piece of plastic wrap. Let rest for 10 to 15 minutes or until the balls puff up a bit. (If you're in a hurry, you can skip this extra resting time and roll out the dough balls right after you form them.)