Remove the top oven rack and place a rack on the bottom 1/3 of the oven. Preheat the oven to 350 degrees F.
Create a parchment paper or aluminum foil collar for the souffle dish; the collar should measure about 2 inches above the top of the dish. Grease the souffle dish and collar separately with 1/4 cup softened butter (see Cook's Note*). Dust the inside of the dish and the collar with 1/4 cup of Parmesan. Reserve any leftover cheese for sprinkling on top of the souffle. Secure the collar to the outside of the dish with butcher's twine.
Melt 1/4 cup of butter in a large saucepan over medium heat. Add the flour and stir constantly for 2 minutes, making sure it doesn't brown. Slowly whisk in the milk until smooth, and continue stirring for about 4 to 5 minutes to form a bechamel sauce; the bechamel sauce should be very thick. Remove from the heat and add the salt, nutmeg, and cayenne pepper.
Beat the 6 egg yolks 1 at a time into the hot bechamel sauce until fully incorporated. Stir in the Gruyere and chives. Set aside (see Cook's Note**).
Place the 8 egg whites in an electric mixer fitted with a whisk attachment. Beat the egg whites until stiff but not dry. Stir 1/4 of the egg whites into the souffle base. Fold in the remaining egg whites, being careful not to over mix. Gently spoon the mixture into the souffle mold and bake for 45 minutes, or when a wooden skewer inserted into the middle of the souffle comes out clean. Minimize opening and closing the oven door. Serve immediately sprinkled with reserved cheese.
*Use a butter wrapper or a little piece of parchment paper to keep your hands clean while smearing the softened butter in the souffle dish. This coating technique is called "chemise." **Up to this point, the souffle base can be made ahead of time.