In a large pot or Dutch oven, prepare the pasta according to package instructions. Drain (don’t rinse) and set it aside.
In the same large pot, melt the butter over medium heat. Once the butter is melted, sprinkle in the flour and whisk together until smooth. It'll be slightly thick. Keep whisking and heating over medium heat for about 3 minutes or until very smooth and the flour taste has dissipated.
Slowly whisk in the room-temperature milk. Once well combined, flavor with mustard, pepper, and salt. Cook on medium-high, until almost simmering, while whisking occasionally for about 2 minutes or until the milk is hot and almost simmering.
Break the cream cheese up into pieces and drop them into the sauce, followed by the mozzarella and the cheddar cheese. Stir until completely melted and very smooth. At this point, the sauce should be slightly thickened but not chunky or stringy.
Remove from the heat and add the pasta. Stir until all the pasta is covered in sauce. Serve warm.