Preheat the oven to 375 degrees F. Butter the bottom and sides of a deep 13-by-9-inch baking dish.
Make the sauce. In a large bowl, mix the ingredients in order, combining completely between additions. Start by adding the flour, paprika, ground mustard, garlic, scallions, hot sauce, cream cheese, a pinch of salt and more grinds of black pepper than normal. Mix this until combined completely. In a liquid measure, mix the heavy cream and pasta water together and add slowly to the bowl while whisking. Add the Cheddar, Gouda, mozzarella and Pecorino Romano cheeses.
Bake. Add the pasta to the bowl of liquid and stir until the liquid seeps into the elbows and everything is distributed evenly. Pour into the prepared dish. Level with the back of your spoon. Toss together the Cheddar and mozzarella in a small bowl and sprinkle evenly over the top. Bake until the edges are bubbly and the top is golden brown, 35 to 40 minutes.