Combine the flours, yeast, and 2¼ teaspoons salt in a large bowl.

Add 3 tablespoons of the olive oil and about 1¼ cups warm water—about the same temperature as the inside of your wrist—and stir with the dough-hook attachment of a stand mixer or by hand.

Knead the dough with the dough-hook attachment of a stand mixer or by hand until it feels smooth and elastic, about 10 minutes.

Grease a large bowl (it’s fine to use the same one you mixed the dough in), add the dough, and turn it over to coat it lightly with oil.

Cover the bowl with a clean kitchen towel or plastic wrap, put it in a warm place, and let the dough rise until more or less doubled in size, about 1½ hours.

Meanwhile, put 2 tablespoons of the olive oil in a medium saucepan over medium heat.

When it’s hot, add the onion and garlic and season with salt and pepper.

Cook, stirring, until the onion begins to soften, about 5 minutes.

Add the tomatoes and adjust the heat so the mixture simmers steadily, and cook, stirring occasionally, until the mixture is thick and saucy, about 30 minutes.

Taste and adjust the seasoning.

Heat the oven to 500°.

Punch down the dough and let it rest for a minute or two.

Generously grease two 13- by 18-inch baking sheets.

Divide the dough in two and put each half on a baking sheet; gently and gradually stretch each into 13- by 18-inch rectangle.

Spread the tomato sauce in an even layer over the dough, and top with the cheese and basil.

Drizzle with a little additional olive oil and sprinkle with more salt and pepper if you like.

Bake until the cheese is bubbly and the edges of the crust are golden brown, 10 to 15 minutes.

Let rest at least 5 minutes at room temperature before cutting and serving.