Preheat the oven to 375°F with a rack in the bottom third. Generously grease a 2 ½-quart straight-sided soufflé dish or saucepan (or 6 to 8 individual soufflé dishes) with butter.

Melt the butter in a medium saucepan over medium heat. Add the flour and cook, whisking constantly, for about 2 minutes. Pour in the milk, whisking to prevent lumps, and bring the mixture to a simmer.

Decrease the heat to medium-low and simmer until the mixture is very thick and smooth, whisking the bottom and edges of the pan often, 2 to 3 minutes.

Remove the pan from the heat and whisk in the Cheddar cheese until completely melted and velvety smooth. Next, add Parmigiano-Reggiano cheese, pimento peppers, sour cream, chives, mustard, 1 ½ teaspoons of the salt, and paprika, whisking until smooth. Stir in the 6 egg yolks. Taste and adjust the seasoning; the flavor should be very robust, cheesy, fairly salty, and positively delicious. Set aside.

Using an electric mixer and a very clean, grease-free bowl, beat the 7 egg whites with the pinch of salt to firm, glossy peaks, meaning that when the whisk is drawn from the whites a peak forms and the tip just barely curls back on itself. Using a flexible spatula or wooden spoon, fold one-quarter of the beaten egg whites into the cheesy base sauce to lighten the mixture, then fold in the rest until just barely incorporated, scooping and gently rolling the two together to preserve as much volume as possible.

Scrape the soufflé mixture into the prepared dish, filling it no more than an inch from the top. (If you have too much of the mixture, see Note below.) Wipe clean the exposed inside edge of the dish and place the soufflé on a rimmed baking sheet. Bake until the soufflé is puffed high, deeply browned on top, and still quite jiggly, 30 to 35 minutes (or 15 to 20 minutes for individual servings). Serve right away, as everyone marvels at its beauty.

Note: If you have too much soufflé mixture to fill your dish just an inch from the top, you have two options: Either butter a smaller dish and bake a second, mini soufflé alongside the big one, or make a foil collar extension for the dish. To do that, tear off a sheet of heavy-duty aluminum foil that is about 4 inches longer than the circumference of the dish. Fold it lengthwise so that you have a 4-to 5-inch-wide strip of foil. Grease one side generously with butter. Wrap the foil strip around the outside of the dish, with the greased side facing in, so that at least 2 inches extends above the rim. Fold the ends together until the foil strip is taut and snug against the outside of the dish. Bake the soufflé as directed.