With steel cutting blade in place, process Parmesan cheese until finely powdered.
Butter a 2-1/2 quart soufflé dish or casserole. Dust with the powdered Parmesan cheese. Set dish aside. (Remainder of Parmesan goes into soufflé mixture. See below.)
Finely shred the Cheddar cheese in the food processor with the shredding disc.Remove and set aside. If using the onion, chop it in the food processor with the steel cutting blade, then sauté the onion in butter in a medium saucepan for 5 minutes.(If omitting onion, melt butter, then proceed.)
Stir in the flour and dry mustard. Cook and stir until smooth and bubbly.
Add the milk all at once. Cook and stir over medium-high heat until the mixture comes to a boil and is smooth and thick. Remove the saucepan from the heat.
Stir in the reserved Cheddar and Parmesan cheeses; stir until melted.
With the plastic mixing blade in place, beat the egg yolks in the mixing container until thick and lemon-colored. Stir a small amount of the hot cheese mixture into the yolks. Return the yolk mixture to the saucepan and blend. Using a mixer, beat the egg whites and cream of tartar in a mixing bowl until stiff but not dry, just until whites no longer slip when bowl is tilted. Gently fold the yolk mixture into the whites. Pour the egg mixture into the prepared baking dish.
Holding a spoon upright, gently make a ring 1-inch deep and 1-inch from the side of the dish. Bake the soufflé in a preheated 350°F. oven 35-40 minutes or until puffy and delicately browned; soufflé should only shake slightly when the oven rack is gently moved back and forth. Serve it immediately.