Finely mince garlic and onion in a food processor or blender. Add the tomatoes and pulse until coarsely pureed. In a large skillet, heat olive oil over high heat. Add the tomato mixture, oregano and salt. When mixture boils, reduce the heat to medium low and simmer for 20 minutes.
While the sauce cooks, boil pasta in salted water until tender. Drain, reserving 1 cup of the cooking water, and place pasta in a large serving bowl. If the sauce appears to be too thick, add a small amount of pasta water to then it, then toss with the pasta. Stir in both cheeses and basil. Cover the dish for a few minutes so the mozzarella begins to melt. Serve immediately.