Ingredients
For the sauce
3 tablespoons plus 1/2 cup extra-virgin olive oil
12 plum or Roma tomatoes (about 3 pounds), peeled and seeded, juices strained and reserved (see headnote)
1 1/2 teaspoons kosher salt, or more to taste
10 cloves garlic
3 large sprigs basil (about 24 leaves plus stems)
1 to 1 1/2 teaspoons crushed red pepper flakes
For assembly
Kosher salt
1/2 cup freshly grated Parmigiano-Reggiano cheese
2 tablespoons unsalted butter, cut into pieces
16 fresh basil leaves, stacked, rolled tightly and cut crosswise into thin ribbons (chiffonade)
Extra-virgin olive oil, for drizzling
Preparation
For the sauce: Heat 3 tablespoons of the oil in a wide saucepan over medium-high heat. Once the oil shimmers, carefully add the tomatoes and salt. Cook for 2 or 3 minutes, stirring, until the tomatoes soften, then reduce the heat to medium; use a potato masher to break down the tomatoes and form a sauce. Add the reserved tomato juices to thin the sauce as needed. Cook for 20 minutes, stirring occasionally. Remove from the heat.
Meanwhile, heat the remaining 1/2 cup of oil in a small saucepan over medium heat. Add the garlic, basil and crushed red pepper flakes. Stir until the basil wilts and the garlic turns light brown, about 5 minutes. Remove from the heat; steep for 5 minutes, then strain through a fine-mesh strainer; discard the solids. Add half of the oil to the tomato sauce. Taste, and add salt as needed.
For assembly: Bring a large pot of water to a boil over medium-high heat. Add a generous pinch of salt.
Meanwhile, warm the sauce in a large saute pan over medium heat.
Add the pasta to the boiling water; cook until it is just shy of tender, about 2 minutes. Drain, reserving at least 1/4 cup of the cooking water. Add the pasta and 1/4 cup of the pasta water to the pan with the tomato sauce; increase the heat to medium-high, shaking the pan to incorporate the sauce and the pasta. Cook for 2 to 3 minutes.
Remove from the heat. Stir in the cheese, butter and basil, tossing gently to incorporate. Divide the pasta among individual wide, shallow bowls. Drizzle with oil, if desired; serve warm.
