Preheat the oven to 350°F and set an oven rack in the middle position.
Use softened butter to grease a 2-quart soufflé or ceramic or glass baking dish. Add ¼ cup of the Parmigiano-Reggiano and roll it around the dish to coat the bottom and sides.
In a small saucepan over medium heat, melt the 3 tablespoons of butter. Add the flour and cook, whisking constantly, for about 1 minute. Add the milk and whisk the mixture until smooth. Continue cooking, whisking constantly, until the mixture thickens and comes to a boil, a few minutes (be sure to scrape the edges of the pan with your whisk, where the mixture thickens first). Remove the pan from the heat and stir in the salt, pepper, and nutmeg. Add the Cheddar and the remaining ¼ cup Parmigiano-Reggiano; stir until the cheeses are melted and the mixture is thick and smooth. Let the mixture cool for 10 minutes, then whisk in the egg yolks and the chives.
In the bowl of an electric mixer fitted with the whisk or beaters, whip the egg whites and cream of tartar on medium-low speed until foamy, about 1 minute. Increase the speed to medium-high and whip until stiff peaks form, about 3 minutes. Add one-third of the milk/cheese mixture to the whites and beat on medium speed until smooth. Add the remaining milk mixture to the whites and, using a rubber spatula, fold until the mixture is uniform (see note).
Pour the mixture into the prepared soufflé or casserole dish (it should come about 1 inch from the top; if you have extra batter, discard or bake it in a separate dish). Place on a baking sheet and slide into the oven. (The baking sheet just makes it easier to move in and out of the oven.) Bake for about 35 minutes, or until puffed and golden brown on top. The soufflé will stay inflated for a while, but it is best to serve it immediately.
Note: Folding is a technique used to mix a light and airy ingredient, like beaten egg whites, into a heavier mixture, like a soufflé base, without deflating the lighter mixture. After combining the two mixtures, use a rubber spatula to cut down to the bottom of the bowl. Pull the spatula toward you, scooping up the contents from the bottom of the bowl, and in one sweeping motion, fold the scooped up portion over the top. Turn the bowl a quarter turn and repeat the motions, scraping the sides of the bowl occasionally, until the ingredients are evenly combined.