Step 1

In a large pot, bring 4 quarts water to a boil. Add 1 tablespoons salt and the cauliflower halves and cook for 5 minutes; begin timing from the moment the cauliflower is added to the pot. Using tongs, transfer the cauliflower to a cutting board; reserve the pot and the water. When the cauliflower is cool enough to handle, chop the florets and stems into pieces slightly smaller than the pasta, discarding the thick, tough core. You should have about 4 cups. Return the water to a boil.

Step 2

In a nonstick 12-inch skillet over medium, cook the oil and garlic, stirring often, until the garlic is golden brown, 2 to 3 minutes. Remove and discard the garlic, then add the cauliflower, pepper flakes and 1/4 teaspoon salt. Increase heat to medium-high and cook, stirring occasionally, until the cauliflower is well browned, 7 to 9 minutes.

Step 3

Meanwhile, add the pasta to the boiling water and cook, stirring occasionally, for 5 minutes. Reserve about 2½ cups cooking water, then drain. Add the pasta and ¼ teaspoon black pepper to the skillet with the cauliflower, then stir in 1 cup of the reserved cooking water. Cook over medium-high, stirring often, until the pasta is al dente, 3 to 5 minutes. If the pan becomes dry before the pasta is done, add another ¼ cup reserved cooking water and continue to cook.

Step 4

When the pasta is al dente, with the skillet still over medium-high, stir in another ¼ cup reserved cooking water. Sprinkle on the pecorino, then stir until the cheese is evenly distributed and melted. Sprinkle on the provolone, then stir until the pasta is glossy and lightly coated with melted cheese, then remove the pan from the heat. If the mixture looks sticky and dry, stir in additional cooking water a few tablespoons at a time until the proper consistency is reached. Taste and season with salt and black pepper. Serve drizzled with additional oil and additional cheese and pepper flakes.