To make the dough:
Dump the water, olive oil, yeast, salt and sugar into a 5-quart container with lid, and using a Danish dough whisk or wooden spoon, stir in the flour. No kneading is necessary. Cover the container, not airtight, and let it rest for about 2 hours on the counter. The dough can be used right away, but it is much easier to handle once it has been thoroughly chilled, about 3 hours. The dough can be stored in the refrigerator for up to 2 weeks.
To make the pizza:
Preheat your oven to the highest setting, which will be 500 to 550 degrees F, with a pizza stone in the bottom third of the oven. Depending on the thickness of your stone it can take between 20 and 45 minutes to preheat.
Cut an 8-ounce piece of dough from your bucket using kitchen shears and quickly form it into a ball, no more than 30 seconds of work. Let it sit on the counter while you gather your toppings.
Roll the ball out into a 1/8-inch-thick round. If the ball is resisting, let it sit for about 5 minutes and it will relax and allow you to roll it more easily.
Sprinkle a pizza peel with flour and place dough on it. Spread the tomato sauce over the dough.
Place the chunks of cheese over the sauce and drizzle with the olive oil. Place in the oven on the preheated pizza stone.
Check for doneness in 8 to 10 minutes; turn the pizza around if it seems to be browning on one side more than the other. It may take up to 5 more minutes.
Let the pizza cool slightly, so the cheese can set a little before cutting with a pizza cutter. This is best done on a cooling rack if you want to keep a crisp bottom crust.