Cook the Spaghetti: Bring a large pot of water and 1 tablespoon of the salt to a boil. Add the spaghetti and cook, stirring occasionally, until it is al dente, 10 to 12 minutes. Before draining, reserve 1½ cups of the pasta cooking water. Then, drain the spaghetti.

Prepare the Sauce: In a cold skillet large enough to hold the pasta (a 12-inch sauté pan is ideal), combine the olive oil and garlic. Cook over medium heat until the garlic begins to sizzle. Then, immediately reduce the heat to medium-low and continue cooking, stirring frequently, until the garlic turns lightly golden, 3 to 4 minutes. Be careful not to let the garlic burn. Remove the skillet from the heat and stir in the red pepper flakes. Add about ½ cup of the reserved pasta water to the skillet to stop the garlic from cooking further.

Combine the Pasta and Sauce: Add the drained pasta to the skillet with the garlic oil, along with ¾ cup of the reserved cooking water and the remaining ¾ teaspoon of salt. Cook over medium-high heat, stirring constantly, until the pasta absorbs some of the sauce, 1 to 2 minutes. Add the remaining reserved cooking water if the pasta seems dry.

Finish and Serve: Right before serving, add the chopped parsley and grated cheese to the pasta. Stir until the pasta is well coated and the sauce becomes creamy. Serve immediately, passing additional cheese at the table.