1. In the bowl of a standing electric mixer fitted with a dough hook, combine the yeast, sugar, and warm water; stir gently to dissolve. Let the mixture stand until the yeast comes alive and starts to foam, about 5 to 10 minutes.
2. Turn the mixer on low and add the salt and 2 tablespoons of olive oil. Add the flour, a little at a time, mixing at the lowest speed until all the flour has been incorporated.
3. When the dough starts to come together, increase the speed to medium; stop the machine periodically to scrape the dough off the hook. Get a feel for the dough by squeezing a small amount together: if it's crumbly, add more water; if it's sticky, add more flour - 1 tablespoon at a time.
4. Mix until the dough gathers into a ball, this should take about 5 minutes.
5. Turn the dough out onto a lightly floured surface and fold it over itself a few times; kneading until it's smooth and elastic.
6. Form the dough into a round and place in a lightly oiled bowl, turn it over to coat. Cover with plastic wrap or a damp towel and let it rise in a warm spot (i.e. over a gas pilot light) until doubled in size, about 1 hour.
7. Once the dough is domed and spongy, turn it out onto a lightly floured counter. Roll and stretch the dough into a cylinder and divide into 3 equal pieces. Cover and let rest for 10 minutes so it will be easier to roll out.
8. Roll the dough out as thinly as possible. I stretched and pulled more than rolled.
9. Prep your Bakerstone Box or preheat your oven to 500 degrees while you make the sauce. Drop the garlic cloves through the feed tube of a food processor as it's running. Pulse until the garlic is finely minced.
10. Add the tomatoes, vinegar, salt, pepper, and oil to the food processor. Pulse to get the desired consistency. (I like mine fairly smooth, but it’s up to you.) Give a taste and add more salt, pepper, or vinegar as needed.
11. Add 2-3 tablespoons of the sauce with the back of a spoon to spread it thinly all over the crust. Top with slices of cheese.
12. Bake for about 7 minutes, on a pizza tray if you're using an oven.
13. Add torn basil leaves to the top, drizzle with olive oil, and salt and pepper. Wait a few minutes before slicing.