Preheat oven to 370 degrees F.
Generously butter a 2-quart soufflé dish with one tablespoon of the butter.
In a 2- to 3-quart pan over medium heat, melt the remaining 4 tablespoons of butter. Add the flour and stir until the mixture is smooth and bubbling. Stir in the milk, cayenne pepper, and salt, and continue stirring until the sauce boils and thickens, about 3 to 4 minutes. Remove from heat.
Add the shredded cheese to the pot with the milk and flour mixture and stir until the cheese is melted. Add the egg yolks (one at a time) and whisk after each addition until the mixture is blended and smooth.
In a bowl, with a mixer on high speed, beat the egg whites together with the cream of tartar until short, stiff, moist peaks form. With a flexible spatula, fold a third of the cheese mixture into the whites just until well blended. Add the remaining cheese mixture and gently fold together until blended.
Scrape the prepared soufflé batter into the buttered dish until it is 3/4 of the way full. Pour any extra batter into buttered ramekins.
Before baking the soufflé run a table knife around the edge of the soufflé batter, about 1-inch from the edge of the dish. This will create a pretty "crown" that puffs up in the middle of the soufflé as it bakes.
Bake the soufflé in the preheated oven until the top is golden brown in color, and the cracks in the top look fairly dry, 25 to 30 minutes for a large soufflé, 15 to 20 minutes for small ones. Serve immediately.