Preheat the oven to 450°F.
In a large bowl, toss the cauliflower florets with 2½ tablespoons of the oil and season with kosher salt and black pepper. Arrange the florets in a single layer on a large rimmed baking sheet and roast them, tossing occasionally, until deeply golden in color, about 25 to 30 minutes. Sprinkle with flaky salt. Remove and set aside.
Bring a large pot of water to a boil. Add 2 tablespoons of the kosher salt and return to a rolling boil. Add the pasta and cook until al dente according to package directions.
When the pasta is about 5 minutes from being done, heat the remaining olive oil in a 12-inch skillet over medium-high heat. Add the capers and cook until they begin to crisp, about 3 minutes. Reduce the heat to medium, add the garlic, and cook until golden, about 2 minutes. Stir in the red pepper flakes and cook for 1 minute more.
Increase the heat to medium-high and add the pasta, roasted cauliflower, lemon zest, and parsley and toss to coat, adding ¼ cup of pasta water or more (up to 1 cup), as needed to loosen up the sauce. Season with kosher salt and black pepper.
Plate in bowls and top with grated cheese.