Place 28-ounce can of crushed tomatoes in fine-mesh strainer set over medium bowl.
Let sit 30 minutes, stirring 3 times to allow juices to drain.
Combine 3/4 cup tomato solids, 1 tablespoon olive oil, and 1/8 teaspoon table salt. (Save remaining solids and juice for another use.).
Place towel or shelf liner beneath stand mixer to prevent wobbling.
Mix flour, water, and table salt in bowl of stand mixer fitted with dough hook on low speed until no patches of dry flour remain, 3 to 4 minutes, occasionally scraping sides and bottom of bowl.
Turn off mixer and let dough rest 20 minutes.
Sprinkle yeast and sugar over dough.
Knead on low speed until fully combined, 1 to 2 minutes, occasionally scraping sides and bottom of bowl.
Increase mixer speed to high and knead until dough is glossy, smooth, and pulls away from sides of bowl, 6 to 10 minutes. (Dough will only pull away from sides while mixer is on. When mixer is off, dough will fall back to sides.).
Using fingers, coat large bowl with 1 tablespoon oil, rubbing excess oil from fingers onto blade of rubber spatula.
Using oiled spatula, transfer dough to bowl and pour 1 tablespoon oil over top.
Flip dough over once so it is well coated with oil; cover tightly with plastic wrap.
Let dough rise at room temperature until nearly tripled in volume and large bubbles have formed, 2 to 2-1/2 hours.
One hour before baking pizza, adjust oven rack to middle position, place pizza stone on rack, and heat oven to 450 degrees Fahrenheit.
Coat rimmed baking sheet with 2 tablespoons oil.
Using rubber spatula, turn dough out onto baking sheet along with any oil in bowl.
Using fingertips, press dough out toward edges of pan, taking care not to tear it. (Dough will not fit snugly into corners. If dough resists stretching, let it relax for 5 to 10 minutes before trying to stretch again.).
Let dough rest in pan until slightly bubbly, 5 to 10 minutes.
Using a dinner fork, poke surface of dough 30 to 40 times; sprinkle with kosher salt.
Bake until spotty brown, 15 to 17 minutes, rotating baking sheet halfway through baking.
Using metal spatula, remove pizza from oven, spread tomato mixture evenly over surface, and sprinkle with shredded mozzarella.
Return pizza to oven and continue to bake until cheese begins to brown in spots, 5 to 10 minutes longer.
