In a very large pot, bring 5 quarts water and 3 tablespoons salt to a boil. Add the pasta and cook until just al dente, about 1 minute less than the package directions. Occasionally give the pasta a stir so it doesn’t stick together. Scoop out and reserve 1 cup of the pasta water before draining the pasta.
Meanwhile, put a large skillet over medium high heat. Add the olive oil, onion, cauliflower, and anchovies and cook, stirring occasionally, until the mixture begins to caramelize, about 3 minutes. Taste and adjust the seasoning, adding salt and black pepper as needed.
Add the red pepper flakes and reserved pasta water and deglaze the pan, scraping with a wooden spoon to get up the browned bits on the bottom of the pan. Add the cooked pasta to the pan, stirring it into the sauce. Cook for 1 minute. Remove from the heat and stir in the Parmesan and extra virgin olive oil. Serve immediately.