Butter a 2-quart round or oval baking dish (such as a 10- x 2-inch round) and coat with 2 ounces of the Gruyère or Comté. The cheese will not cover the inside of the baking dish completely. There will be gaps. For beating the egg whites, you can use a stand mixer, an electric hand mixer, or a manual rotary egg-beater.
It's best to use a metal bowl. Wash the bowl and beater in hot soapy water; rinse well and dry. Bowl and beater must be grease-free. Adjust an oven rack to the lower third position and preheat the oven to 400 degrees.
Put 4 tablespoons of the butter into a large skillet (12-inch) over medium heat. When hot, add the garlic and cook about 15 seconds, stirring with a wooden spoon. Add the corn, red bell pepper, and jalapeño and cook, stirring occasionally, until the peppers and corn are partly tender, 2 to 3 minutes. Take the pan off the heat and add 1/4 teaspoon salt and 1/4 teaspoon pepper.
Combine the milk and half and half in a medium heavy saucepan (3-quart) and bring to he boil over medium heat. Watch carefully so the liquid doesn’t overflow the pan. Keep the liquid hot on low heat. (Or, just heat the milk and half and half in a 2-cup heatproof measure (such as Pyrex®) with pouring spout for 2 minutes in a microwave oven).
In another medium saucepan (3-quart), melt the remaining 4 tablespoons butter over medium heat. Add the flour and cook, stirring constantly with a wooden spoon, for 2 minutes. Take the pan off the heat, wait a few seconds for the bubbling to subside, and pour in the hot liquid all at once.
Whisk vigorously to make a smooth béchamel sauce. Return the pan to medium heat and bring to the boil, whisking constantly. Cook at the boil for 2 minutes until very thick. Off heat, whisk in 1/2 teaspoon salt and 1/4 teaspoon pepper. Whisk the egg yolks in one or two at a time.
Transfer the soufflé base to a large bowl. Stir it occasionally with a heatproof flexible spatula until tepid. The base must not be hot when you fold in the beaten egg whites and cheese.
For the egg whites. Start beating on medium speed, and beat until the whites are frothy, about 1 minute. Add the cream of tartar or lemon juice—the acid stabilizes the whites—and continue beating until the whites form moist-looking peaks that stand up straight or curl just a tiny bit at their tips.
Slowly tilt the bowl to make sure the whites are staying put. If they slide around, beat a few seconds longer. As soon as you can turn the bowl upside down with the whites staying put, they’re beaten just right. Do not overbeat or the whites will be too stiff.
Whisk about 1/4 of the whites into the tepid soufflé base just to lighten it. Gently fold in the remaining whites in two additions, followed by the cooled corn and peppers and 3 ounces of Gruyère or Comté. Be gentle to maintain as much air in the whites as possible.
Scrape the soufflé batter into the prepared dish—it will be almost full. Sprinkle the remaining ounce of cheese on top and place the soufflé in the oven. Bake 30 minutes, until the soufflé has puffed up to double its volume and is well-browned on the top and sides. A wooden skewer plunged into the center of the soufflé should come out clean.
Serve as soon as you can. But do not worry. The soufflé will not collapse. It will just settle a bit. Refrigerate leftovers. To reheat, cut the cold soufflé into slices and reheat each serving on a plate for 1 1/2 to 2 minutes in a microwave oven. Delicious!