Prepare the Béchamel Sauce:

Preheat your oven to 375°F (190°C).
In a medium saucepan, melt 2 tablespoons of butter over medium heat. Stir in the flour and cook for about 2 minutes without letting it brown.
Gradually whisk in the milk until the mixture is smooth. Cook, stirring constantly, until the sauce thickens.

Mix the Ingredients:

Stir the egg yolks into the béchamel sauce one at a time, then add the Gruyère, half of the Parmesan, nutmeg, salt, and pepper.
In a separate bowl, beat the egg whites with a pinch of salt until stiff peaks form.
Gently fold the egg whites into the cheese mixture.

Bake the Soufflé:

Grease a 6-cup soufflé dish with butter and sprinkle with the remaining Parmesan cheese.
Pour the soufflé mixture into the dish and smooth the top.
Bake in the preheated oven for about 25-30 minutes, or until the soufflé is puffed and golden brown. Do not open the oven door during the first 20 minutes of baking.

Serve Immediately:

Serve the soufflé immediately, as it will start to deflate once it's removed from the oven.