Preheat the oven to 400 degrees F.
Butter 6 (7 1/2 by 3 1/2-inch ramekins. Sprinkle with some grated Parmesan, and coat all sides and the base of the dish. Pour out the excess cheese into the next ramekin to coat. Continue coating the ramekins with the cheese and set aside.
Heat the milk in a small saucepan over low heat.
In another medium saucepan, melt 2 tablespoons of butter over medium heat. Whisk in the flour and cook out raw flour taste, about 2 minutes. Whisk in the hot milk and bring to a boil, then reduce the heat to a simmer. Remove the pan from heat, add 3/4 cup of grated cheese and season, to taste, with salt and pepper (watch salt levels because of the cheese). Pour the mixture into a large bowl and set aside to cool to room temperature.
In a stand mixer, whip the egg whites to stiff peaks.
Whisk the yolks into the cooled bechamel. Fold 1/3 of whipped egg whites into the bechamel to lighten the base. Gently fold in the remaining whipped egg whites in order to keep the incorporated air. Carefully fill the prepared ramekins and arrange them on a baking sheet. Bake until risen and golden on top, about 20 minutes. The center will have just a slight jiggle when ready. Remove the ramekins from the oven and serve immediately.
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