Meanwhile, in a large deep skillet, cook the garlic and chili peppers in the olive oil over medium-low heat until the garlic starts sizzling with little bubbles around it, 1 to 2 minutes. Add the basil leaves and sizzle until fragrant, about 15 seconds. Increase the heat to medium-high, add the tomatoes, season with salt, and spoon over a ladle (½ cup) of pasta cooking water. Cook until the tomatoes are soft but not quite melted into the sauce, about 5 minutes. Add more water as needed to keep the sauce glossy.

Transfer the pasta to the sauté pan along with a ladle (½ cup) of pasta cooking water and cook until the pasta is al dente. Remove from the heat and cook until the pasta is al dente. Remove from the heat and stir in the Parmigiano-Reggiano and Pecorino Romano, tossing with the passion of an Italian until the sauce is thick and clings to each pasta strand, adding more pasta water as needed to keep the sauce very juicy.

Twirl the pasta into two nests and serve on a single plate garnished with a scrunch of pepper and fresh basil leaves.