Cook the pasta as per package directions in a large pot until al dente. Reserve 1 cup of the pasta water for later. Drain and set aside.
In the same large pot over medium heat, add the remaining ingredients all to the pot. Stir to combine and heat until the cheeses are all melted and everything is incorporated, approximately 5-7 minutes.
Season the cheese sauce to taste with cheese, red pepper flakes, and salt. If too thin, continue cooking to thicken. If too thick, add a little of the pasta water.
Add the pasta back into the pot and stir to coat. Add a little of the reserved pasta water if needed to help the sauce mix with the pasta. Serve immediately.