Begin by shredding the Gruyere and either grating or finely shredding the Parmigiano Reggiano. Set the cheeses aside.
In a small saucepan, heat the milk over medium heat until it reaches a simmer. Remove the saucepan from the heat while you proceed to make the roux for the béchamel base.
In a separate medium saucepan, melt the butter over medium heat. Then add the flour to create a roux and whisk constantly for about two minutes, until the mixture is foamy and pale yellow with a pleasant toasty aroma.
Gradually add the hot milk, whisking continuously, and bring the mixture to a boil.
Cook, continuing to whisk, for about 5 minutes or until the mixture is noticeably thickened.
Remove the saucepan from the heat and add the freshly shredded Gruyere and Parmigiano Reggiano. Whisk until the cheeses are melted and the sauce is smooth.
Stir in the freshly grated nutmeg and season to taste with salt and pepper.