Bring a large pot of water to a boil for the pasta. (Heads up: If you can find pici, it takes about 20 minutes to boil to al dente. Bucatini will take only about 7 minutes, so time your sauce accordingly: It'll take just under 30 minutes.)
In a large skillet, heat the olive oil, four turns of the pan, over medium-low. Add the garlic and cook, partially covered, stirring occasionally, until the garlic is soft, about 5 minutes. Add the tomatoes to the pan. Increase the heat to medium-high; season the tomatoes with salt. Cover and cook, occasionally shaking the pan, until the tomatoes soften, about 20 minutes. During the last 5 minutes of cooking, coarsely tear the basil and add it to the sauce.
Salt the boiling water, add the pasta, and cook to almost al dente. Reserve 1/2 cup of the starchy cooking water, then drain the pasta. Return the pasta to the hot pot. Add the butter, the reserved starchy cooking water and half the sauce; toss, over medium, to combine. Add a couple handfuls cheese, tossing with the pasta to coat. Serve the pasta in shallow bowls topped with the remaining sauce. Pass the crushed red pepper and more cheese at the table for topping.