Mix the dough. In a medium bowl, whisk the flour, yeast and salt. Add the water and 1 tbsp oil and stir until no dry flour remains. Rest 10 minutes, then knead on the counter for 3–4 minutes, until the dough is smooth and springs back slowly when poked.
Let it rise. Place the dough in a lightly oiled bowl, cover, and let it rise at warm room temperature for 1½–2 hours, until roughly doubled and puffy. For the overnight version, use ½ tsp yeast and refrigerate the covered dough for 12–72 hours, then let it sit at room temperature for 1–2 hours before shaping.
Prep the broiler and toppings. Set an oven rack about 6 inches below the broiler and heat the broiler on high for 10 minutes. Stir the crushed tomatoes with a pinch of salt. Pat the mozzarella pieces dry with paper towels.
Shape. On a lightly floured counter, press the dough into a flat disk with your fingertips, then stretch it into a 10- to 11-inch round, leaving the outer ½ inch slightly thicker. If it keeps shrinking back, cover it and rest 5 minutes, then stretch again.
Heat the skillet. Add 1 tbsp oil to the cast iron skillet and set it over medium heat for about 3 minutes, until the oil shimmers but doesn't smoke.
Add the dough and top it fast. Take the skillet off the heat. Drape the dough over your knuckles and lay it in the pan away from you, then nudge the edges flat with a spatula (the pan is very hot). Spread the tomatoes to within ½ inch of the edge, scatter the mozzarella and Parmesan, and return the skillet to medium heat.
Cook the bottom. Cook for 4–6 minutes, popping any large bubbles with a fork, until the underside is golden brown with darker spots when you lift an edge with the spatula.
Broil the top. Move the skillet to the rack under the broiler and broil for 3–5 minutes, until the rim is puffed and browned with a few charred spots and the cheese is melted and bubbling. Check after 2 minutes; broilers vary a lot. No broiler? Bake at 500°F (260°C) on the upper rack for 6–8 minutes instead.
Finish and slice. With an oven mitt on the handle, set the skillet on the stove. Scatter the basil, drizzle with 1 tsp oil, and rest 2 minutes. Slide the pizza onto a cutting board and cut into 6 wedges with kitchen shears or a pizza wheel.