Set a pizza stone on the lower rack of the oven, remove the other racks, and preheat to 500 degrees F.
Using a food mill, grind the tomatoes into a bowl (or pulse in a food processor until coarsely pureed.) Season with sea salt and set aside.
Very lightly dust a work surface with some flour. Fill a medium bowl with flour and spread it out into a wide even layer. Drop the dough onto the flour in the bowl, and flip it over once in the bowl to lightly coat it. On the work surface, using your fingertips, press the dough all over to flatten it. Flip it over and press all over the surface again. Flip it one more time and repeat. Press the dough with your fingers repeatedly on one side with one hand, and with the other hand, slowly turn the dough clockwise on the work surface. Continue poking and rotating the dough to flatten and stretch the dough.
Pick the dough up and toss it back and forth between the palms of your hands until it begins to stretch, forms a rough disc, and gets thinner. Drape the dough over your hands, tuck your fingers in, and gently pull your knuckles toward the outer edge of the dough. Continue, rotating the dough as you go, until it is about 10 inches in diameter. Work slowly, allowing the weight of the dough itself to stretch and do not force it or it will tear and can make the crust too chewy.
Dust the work surface with a little more flour and set the crust back down. Press it with your fingers and pull it gently at the edges to even out the circle. Using a small ladle, evenly spread about 1/4 cup of the tomato sauce over the surface of the dough, leaving a 1/2-inch border around the edge. Scatter the mozzarella pieces from the center out, leaving space between the cheese. Tear the basil leaves in half and scatter the pieces over the pizza.
Lightly flour a pizza peel and set it on the work surface at the edge of the pizza crust closest to you. Run your fingers around the outside parameter of the crust to loosen any edges that might be stuck (and make sure the dough is not stuck to the counter anywhere in the center). Gently grab the dough at the 5 o'clock and 7 o'clock positions and in one, quick motion, pull the pizza onto the peel. Use your fingers to pull and straighten the crust until round.
Open the oven door. Set the front of the peel on the back edge of the pizza stone. Shake the peel very lightly to make sure the dough is not stuck and will slide. Holding the peel at a 45-degree angle, jiggle it until the pizza begins to slide off; once it reaches the edge of the peel, shake it lightly a few times, and then pull the peel out in one swift motion to drop the pizza onto the hot stone. Immediately close the oven door and cook until the crust begins to char on the bottom and edges and cheese is melted and bubbly, 8 to 10 minutes.
Use the peel to remove the pizza; slide it onto a cutting board. Slice with a pizza wheel and serve immediately.
Katie's Pizza Dough:
Put the yeast in a measuring cup and dissolve it in about 1/2 cup of the water. In a large bowl, dissolve the salt in the remaining water.
Add about 1/2 cup of the flour to the salt-water solution and add the yeast mixture. Use your fingers to mix well. Add the remaining flour and mix with your hands until it forms a dough. Transfer the dough to a floured countertop.
To knead the dough, grab the sides with your fingers and pull it up and over itself into the center. Using the balls of your hands, press the dough firmly in the center. Repeat, pulling the dough from the outer sides inward, Press it fast from the edge inwards. Working quickly will help prevent it from sticking. Continue kneading the dough until it is very smooth to the touch, and springs back when pressed, 15 to 20 minutes. Shape the dough into a ball, place it in a clean bowl, and cover with plastic wrap.
Let the dough rise for 2 hours, and then divide into four 9-ounce portions. Form the portions into balls. Cover the dough with a towel and let rise at room temperature for 6 hours.