Bring a large pot of salted water to a boil over high heat. Add the pasta and stir.
While the pasta cooks, heat 1/4 cup olive oil in a large skillet over medium heat. Add the garlic and cook until the slivers are golden brown and crisp, then add the chiles and cook for about 30 seconds. Raise the heat to high and add the tomatoes. Cook until the tomatoes soften and the juices thicken, about 3 minutes. Add the basil and season with salt and pepper.
When the pasta is al dente, scoop out about 1/2 cup of the pasta cooking water, then drain the pasta. Return the pasta to the warm pot off the heat. Add the sauce and mix well. Add some of the reserved cooking water if the pasta seems dry. Transfer to a warm serving bowl and grate Parmesan over the top, to taste. Drizzle with a little more olive oil. Serve immediately.