In a large bowl, mix the yeast with 1/2 cup of the warm water and the sugar and let stand until foamy, about 5 minutes. Add the remaining 1 1/2 cups of warm water, the 4 cups of flour and the kosher salt and stir until a soft dough forms. Turn the dough out onto a well-floured work surface and knead, adding flour as necessary until a silky, but soft dough forms. Use a pastry scraper to help knead the dough. Transfer the dough to a lightly oiled bowl and brush all over with olive oil. Cover the bowl with plastic wrap and refrigerate overnight or for up to 3 days.
Transfer the dough to a lightly floured surface; punch down and divide into 4 pieces. Form each piece into a ball. Rub each ball with oil and transfer to a baking sheet. Cover the balls loosely with plastic wrap and let rise in a draft-free place for 1 hour.
Meanwhile, set a pizza stone in the oven and preheat the oven to 500 allowing at least 45 minutes for the stone to heat. Pass the tomatoes through a food mill set over a medium bowl or pulse them in a food processor until coarsely chopped. Stir in the oregano and 1 tablespoon of the olive oil and season generously with salt and pepper.
On a lightly floured surface, stretch one ball of dough into a 13-inch round; transfer to a floured pizza peel, adding flour where the dough sticks. Spread 1/4 cup of the tomato sauce over the dough to within 1 inch of the edge. Spread one-fourth of the cheese over the pizza and drizzle with 1 tablespoon of oil. Season with sea salt and pepper and slide the pizza onto the stone. Bake until the bottom is charred and the cheese is melted, about 8 minutes. Scatter one-fourth of the basil on top and let stand for 3 minutes before serving. Repeat with the remaining dough and toppings.