Prepare the soufflé dish. Preheat the oven to 400°F. Generously butter a 6-cup soufflé dish with 1 tablespoon of butter and coat the inside with some of the grated Parmesan.
To make a foil collar, fold a long piece of foil in half lengthwise, butter one side, and wrap it around the outside of the dish with the buttered side facing inward.
Make the roux. Melt the remaining 3 tablespoons of butter in a saucepan over medium heat. Stir in the flour and cook for about 2 minutes, stirring constantly, until the mixture is foamy but not browned.
Add the milk and seasonings. Remove the pan from the heat and add the warm milk all at once and continue whisking until smooth. Stir in the paprika, nutmeg, salt, and white pepper. Return the pan to medium heat and whisk for about 1 minute, until the mixture is very thick.
Whisk in the egg yolks. Remove the saucepan from the heat, then whisk in the egg yolks one at a time until fully incorporated. Set aside.
Whip the egg whites. Using a stand mixer or hand mixer, add the egg whites to a wide bowl and beat with a whisk attachment, starting on low speed and gradually increasing to medium-high.
Continue beating until the whites are glossy and hold stiff peaks when you lift the whisk. Be careful not to overbeat and don’t let the egg white sit for too long.
Fold the soufflé mixture together. Stir a large spoonful of whipped egg whites into the egg base to lighten it. Gently fold in about half of the remaining egg whites using a rubber spatula, cutting down through the center and lifting the mixture up and over itself.
Add the Gruyère and continue folding in the remaining egg whites until just combined. A few small white streaks are fine, just be sure to avoid over-mixing.
Fill the dish and bake. Gently pour the mixture into the prepared soufflé dish. Sprinkle with the remaining Parmesan on top. Place the dish on a baking sheet and transfer it to the oven. Immediately reduce the oven temperature to 375°F and bake for 30 minutes without opening the oven door.
Serve immediately. The soufflé should rise above the rim and develop a golden top. Serve immediately, using two spoons to scoop portions from the top down through the airy center.