In a medium saucepan placed over medium heat, melt the butter. Add the flour and cook, whisking constantly, until the flour turns light brown and emits a nutty aroma.
Slowly add the milk, whisking constantly. Once all of the milk has been incorporated, whisk in the nutmeg, and allow the mixture to come to a simmer. Once simmering, cook for 2 minutes, whisking constantly.
Remove from the heat, and add the shredded provolone a handful at a time, stirring until it melts. Stir in the Parmesan cheese and season with salt and pepper.