Bring a large pot of salted water to a boil over high heat. Cook the pasta 2 minutes less than the packaging suggests, about 8 minutes. Drain well, reserving 1 cup of the pasta water.
Using the large holes on a box grater, grate the flesh of the tomatoes, stopping when you are left with just the skin. Heat a large straight-sided skillet over medium-high heat and add the oil. Once the oil is hot, add the garlic and cook, stirring constantly, until the garlic is fragrant and beginning to brown, about 2 minutes. Add the grated tomatoes, whole basil and 3/4 teaspoon salt to the hot oil. Reduce the heat to medium-low and cook for 5 minutes.
Remove the garlic and basil stems from the pot and stir in the ricotta. Add the pasta followed by the Parmigiano-Reggiano. Stir to combine and coat everything in the sauce.
Cook for another 2 minutes to finish cooking the pasta and thicken the sauce slightly, adding pasta water if the sauce ever looks too dry. Stir in the chopped basil and serve topped with additional ricotta and Parmigiano-Reggiano.