Combine the tomato puree, basil, kosher salt and 2 tablespoons of the olive oil in a small bowl. Allow to sit for 1 hour to let the flavors marry. Remove the pizza dough from the refrigerator and bring to room temperature for an hour as well.
Preheat the oven to 400 degrees F.
Drizzle the remaining 2 tablespoons olive oil on a 9-by-13-inch rimmed baking sheet. Stretch the dough in the oil, flipping it back and forth and stretching it to fit the baking sheet. Use your fingers to make a few dimples in the dough. Spread the sauce evenly over the top all the way to the edges. Bake until bubbly and golden brown, 20 to 25 minutes.
Allow to cool for 5 minutes on the baking sheet, then remove to a wire rack to cool completely. Drizzle with 1 tablespoon olive oil and sprinkle with the ricotta salata, oregano and flake salt. Cut into squares and serve warm or at room temperature.