Preheat the oven to 400 degrees Fahrenheit (200 degrees Celsius) for a more browned soufflé with a somewhat looser middle, or 375 degrees Fahrenheit (190 degrees Celsius) for a more softly cooked soufflé that sets all the way through. Place the oven rack in the bottom of the oven. Softened butter should be used to grease the inside of a 48-ounce ramekin. Add some grated Parmigiano-Reggiano and rotate the ramekin all around so the cheese attaches to every area of the greased surface; if you don't have enough, add more cheese or discard any excess. Remove the prepared ramekin from the refrigerator and wipe the rim of the soufflé dish.
Melt 3 tablespoons butter in a small saucepan over medium-high heat (do not allow it to brown). Whisk in the flour to make a paste. Cook, stirring constantly, until the raw flour smell is gone, about 1 minute. Add milk in a thin, steady stream, or a couple of teaspoons at a time, whisking well and getting into all corners of the pan to retain a lump-free texture, while whisking constantly. When all of the milk is added, the sauce will thicken at first, then thin out.
Heat, stirring constantly, until the sauce comes to a simmer and thickens slightly. Reduce heat to low and cook, stirring and scraping the sides and bottom of the pan, for 3 minutes, or until sauce has thickened beautifully. Season with salt and pepper to taste. Allow to cool slightly before transferring the béchamel sauce to a large heatproof mixing bowl.
In a béchamel sauce, whisk together Dijon mustard (if used) and cayenne or hot sauce (if using). Then, one at a time, mix in the egg yolks until well combined, while constantly whisking. Set aside the soufflé base.
Combine egg whites and cream of tartar (if using) in a large mixing bowl and beat until firm, glossy peaks form, using a French whisk, electric hand blender, or stand mixer fitted with whisk attachment.
1/3 of the beaten egg whites should be added to the béchamel foundation and whisked until well incorporated and the soufflé base has a looser consistency. Gruyère should be added at this point. Gently fold the remaining beaten egg whites into the soufflé foundation with a silicon spatula until barely incorporated.
Take the soufflé dish out of the fridge and place it on a rimmed baking sheet. Scrape soufflé batter into prepared baking dish, filling it to the inside ridge but not all the way to the top (discard any excess). If desired, smooth and level the surface of the soufflé batter with an offset or other spatula.
Transfer soufflé to oven and bake until well risen and nicely browned on top, about 30 minutes at 400°F for less set, 35 minutes at 400°F for more set, and 35-40 minutes at 375°F for fully set, about 30 minutes at 400°F for less set, 35 minutes at 400°F for more set, and 35-40 minutes at 375°F for fully set.
Transfer the soufflé to the table as soon as possible before it deflates too much, scooping parts onto each diner's plate.