Place a rack in the center of the oven, and preheat the oven to 350 degrees.
Rub the butter on the bottom only of a 2-quart (8-cup) souffle or baking dish and set aside.
Place the milk in a medium saucepan over medium heat and bring to a boil. Pull the pan off the heat, and whisk in the grits, a little at a time, until the grits are incorporated. Place the pan back over medium heat, and let the grits come to a simmer. Stir and cook the grits until smooth and thickened, about 8 to 10 minutes.
Turn off the heat, and stir in the salt, pepper, cayenne or nutmeg, cheese, and butter until smooth. Set aside.
Crack the eggs, placing the yolks in a medium bowl and the whites in a large bowl. Beat the egg yolks lightly with a fork, and add a couple tablespoons of the warm grits to the yolks to bring up their temperature. Stir to combine. Repeat this process two or three times, adding a couple tablespoons grits to the yolks, and then turn the yolk mixture into the pan of grits, stirring to combine well. Set aside.
Beat the egg whites on high speed with an electric mixer until stiff peaks form, 2 to 3 minutes. Pour the grits mixture alongside the egg whites in the bowl and fold gently to combine. Turn the mixture into the prepared dish, and place the dish in the oven.
Bake until the souffle rises and is golden brown on top, about 40 to 45 minutes. Serve at once.