Coat interior of a 10- to 11-cup soufflé dish with 1 1/2 teaspoons butter. Add parmesan cheese; tilt dish to coat.
Separate eggs; put yolks in a small bowl and all whites in a large mixer bowl. Add cream of tartar to whites.
Melt remaining butter in a 10- to 12-inch frying pan over medium-high heat. Add flour and stir until bubbling.
Remove from heat. Whisk milk into the mixture. Add salt, pepper, cayenne, and ground nutmeg. Return to high heat and stir until mixture boils, about 2 minutes; sauce will be very thick. Set aside about 2 tablespoons gruyère cheese, then add remainder to sauce and stir until melted. Remove from heat. Dump in egg yolks and mix well.
With a mixer on high speed, beat egg whites just until they hold soft but distinct peaks. Put a big spoonful of whites into the sauce and mix well.
Scrape sauce into whites, and with a flexible spatula, fold ingredients together; the mixture can look streaky.
Scrape mixture into soufflé dish. With the tip of the spatula, draw a circle on top of the soufflé, then sprinkle with reserved gruyère cheese.
Set on the center rack of a 400° oven. Bake until browned and center jiggles slightly when dish is gently shaken (for a saucy soufflé), about 25 minutes, or for a drier soufflé, about 10 minutes more. Serve at once.