Preheat your oven to 375°F (190°C) with a rack in the lower third. Generously butter the bottom and sides of a 1.5-quart (6-cup) soufflé dish. For extra insurance, you can dust the buttered dish with a tablespoon of grated Parmigiano-Reggiano, tapping out the excess. This creates a flavorful crust that helps the soufflé climb.
Make the béchamel base (panade). In a medium saucepan over medium heat, melt the 3 tablespoons of butter. Whisk in the flour and cook, stirring constantly, for about 2 minutes until it smells nutty but is not browned. Trust me, cooking the flour is crucial to remove the raw taste. Whisk in the cold milk all at once. Continue whisking vigorously until the mixture thickens and comes to a simmer. Cook for 1 more minute, then remove from heat. Whisk in the Dijon, nutmeg, cayenne, salt, and pepper. Let cool for 5 minutes, then whisk in the egg yolks one at a time until fully incorporated. Finally, stir in the grated Gruyère until melted and smooth. Set aside.
Whip the egg whites. In a scrupulously clean, dry bowl, use a hand mixer or stand mixer to beat the egg whites and cream of tartar on medium speed until foamy. Increase speed to high and beat until stiff, glossy peaks form. Tip from me: The peaks should hold their shape when you lift the beaters, but the mixture should still look moist and smooth, not dry or grainy.
Fold the mixtures together. Add about one-quarter of the whipped egg whites to the slightly cooled cheese base. Stir vigorously to lighten the base. Now, gently fold in the remaining egg whites in two additions using a large rubber spatula. Believe me, it's okay if a few white streaks remain; over-mixing is the enemy of a tall soufflé. Pour the batter into the prepared dish and smooth the top.
Bake and serve immediately. Run your thumb around the inside edge of the dish to create a 'moat' – this helps the soufflé rise straight up. Place it in the oven and immediately reduce the temperature to 350°F (175°C). Bake for 25-28 minutes without opening the oven door. It's done when the top is deep golden brown, the center jiggles slightly, and an instant-read thermometer inserted into the center reads 160°F (71°C). Serve immediately straight from the oven.