Place water, sugar and yeast in the bowl of a food processor. Sprinkle yeast over the water and pulse once or twice to mix. Let rest 5 minutes to dissolve.
Add olive oil and pulse a couple of times to blend.
Add 2 cups flour and pulse a few times to mix the flour with the liquid ingredients. Add the rest of the flour until dough forms a ball and pulls away from the sides of the bowl. Continue processing until dough pulls together and develops elasticity.
Remove to a floured surface and finish kneading by hand, adding more flour but not too much so the dough remains light.
Place ball of dough in an oiled bowl, turning dough to cover with the oil. Let rise until doubles, about 2 hours.
If you need more time then either deflate the dough and let it rise once more or place the bowl with the dough in the fridge until you are ready. Bring it to room temperature before proceeding with rolling the dough and shaping the pizza.
When ready to proceed turn the dough onto a lightly floured surface and cut into 4 pieces. Keep the extra pieces covered while you work with each one. Either roll or stretch the dough into a rough circle (it doesn’t have to be a perfect round, rougher shaped pizza are very attractive). Keep the edges slightly thicker than the center. I let the dough rest between stretches, if the gluten is relaxed it is more likely to stretch and cooperate.
Cut parchment paper into squares that the pizzas can fit on. Lay the dough round on the parchment paper, it makes it easy to transfer it into the oven. Brush it with olive oil and then proceed with the topping of your choice.
Use a pizza peel or a wide spatula that will fit the entire pizza with the parchment to lift the pizza and then slide the dough still on the parchment sheet onto the pizza stone in the oven.
Place a baking stone in the oven about 4 inches below the broiler. Preheat oven at broil setting for 30 minutes. When ready, slide the pizza with the parchment onto the hot stone and broil until the topping is bubbly and the crust is crisp and charred in spots.
See below for topping instructions.