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.
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