Boil the water. Bring 4 quarts of water and 1 tablespoon kosher salt to a rolling boil in a large pot. While it heats, mince the garlic, grate the Parmesan, tear the basil and set the cream on the counter.

Cook the pasta. Add the pasta, stir for the first 30 seconds so it doesn't stick, and cook 1 minute less than the package's al dente time. Before draining, scoop out 1 cup (240 ml) of the starchy cooking water.

Bloom the garlic. As soon as the pasta goes in, heat the olive oil and butter in a 12-inch skillet over medium heat until the butter foams. Add the garlic and pepper flakes and stir for 30 to 60 seconds, until fragrant and just turning pale gold at the edges. Don't let it brown.

Toast the tomato paste. Stir in the tomato paste and cook for about 1 minute, until it darkens from bright red to brick red and starts to stick lightly to the pan.

Simmer the tomatoes. Add the crushed tomatoes, 1/2 teaspoon salt and the black pepper. Bring to a lively simmer and cook for 5 minutes, stirring now and then, until the sauce thickens and a spoon dragged through it leaves a trail that fills in slowly.

Add the cream gently. Turn the heat to low and wait about 30 seconds for the bubbling to calm down. Pour in the cream in a slow stream while stirring. Heat for 1 to 2 minutes, until the sauce is steaming with only small bubbles at the edges. It should look evenly orange-pink, with no white streaks or flecks. Don't let it boil hard from here on.

Toss the pasta. Add the drained pasta and 1/4 cup (60 ml) of the reserved water to the skillet. Toss with tongs over low heat for about 1 minute, until every piece is coated.

Finish off the heat. Take the pan off the burner. Add the Parmesan in two handfuls, tossing between each, and loosen with more pasta water, 1 to 2 tablespoons at a time, until the sauce is glossy and clings loosely to the pasta. Fold in the torn basil, taste and add salt if needed.

Serve right away with extra Parmesan and a few small basil leaves on top.