Shuck the Oysters (10), rinse them clean to get rid of any impurities, gently dab dry, then set aside. In a pot over medium heat, melt the Unsalted Butter (2 tablespoon). Add the All-Purpose Flour (2 tablespoon) and cook for a minute.
Whisk in the Milk (1 cup) and Dry White Wine (1/4 cup). Continue whisking until the mixture comes to a simmer and has fully thickened, then keep cooking for 56 minutes until reduced slightly and alcohol has evaporated.
Turn off the heat. Add Shredded White Cheddar Cheese (3/4 cup), Gruyère Cheese (3/4 cup), Garlic (1 clove), Sea Salt (1/3 teaspoon), Ground Black Pepper (1/4 teaspoon), Ground Nutmeg (1/8 teaspoon), and Ground White Pepper (1/8 teaspoon).
Stir with a fork until the cheese has fully melted. Taste and reseason with sea salt if needed.
Preheat the top broiler on high. In a shallow oven-proof skillet, spread half of the cheese sauce on the bottom, then arrange the oysters evenly and cover with the rest of the cheese sauce.
Top with Parmigiano-Reggiano (to taste) to entirely cover the surface, then scatter a few extra nubs of unsalted butter here and there. Bake on the middle rack of the oven for 13 to 15 minutes, until it's bubbly and golden browned.
Grate another generous pinch of fresh nutmeg over the top. Don't be shy with the nutmeg! Serve immediately with Sourdough Bread (to taste).