Preheat the oven to 400° and brush seven 1-cup ramekins with butter. Lightly coat the ramekins with 2 tablespoons of the Parmigiano-Reggiano and set them on a sturdy baking sheet.
In a medium saucepan, melt the 4 tablespoons of butter. Whisk in the flour and cook over moderate heat for 1 minute. Whisk in the milk and cook over moderately low heat until smooth and very thick, about 2 minutes. Stir in the salt and cayenne. Off the heat, whisk in the egg yolks. Let cool slightly. Transfer to a large bowl and stir in the Gruyère.
In a large bowl, using an electric mixer, beat the egg whites with the cream of tartar at medium-high speed until frothy. Increase the speed to high and beat until firm peaks form. With a rubber spatula, fold the egg whites into the soufflé base until no streaks of white remain.
Spoon the soufflé mixture into the ramekins, filling them to 1/2 inch below the rim. Run your thumb inside the rim of each ramekin to help the soufflés rise evenly. Sprinkle the remaining Parmigiano-Reggiano on top and bake in the bottom third of the oven until the soufflés are puffed and golden brown, about 20 minutes. Serve immediately.
The recipe can be prepared through Step 2 and refrigerated overnight. The soufflés can be assembled up to 1 hour ahead. The cooked soufflés can be reheated in the oven.
Dry, minerally Alsatian Riesling.
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