Bring a large pot of water to a boil, then sprinkle in 1 tablespoon kosher salt. Add the pasta and cook to al dente, about 8 minutes, then drain. Return to the pot and add the butter. Stir until the butter has melted and coated the pasta.
In a separate bowl, whisk together the milk, eggs, salt, mustard, hot sauce, and pepper. Stir into the pasta and add the cheese.
Set over low heat and stir until creamy, about 3 minutes. Serve.