Preheat oven to 400° F. Butter a 2- to 2 ½-quart soufflé or baking dish; set aside. Place the bread in a shallow bowl and pour the milk over it. Set aside until the liquid is absorbed. Meanwhile, melt the 2 tablespoons of butter in a medium skillet over medium-low heat. Add the onion and sauté until tender, about 15 minutes. Remove from heat. Mash the bread with your hands or a spoon and stir in the onions. In a medium bowl, whisk together the egg yolks, cheese, salt, pepper, nutmeg, and cayenne. Pour the mixture over the bread, stirring until well combined. In a separate bowl, beat the egg whites until stiff peaks form, then gently fold them into the bread mixture in 3 additions. Transfer to the soufflé dish and bake in oven until golden and firm, 45 to 50 minutes. Serve with salad greens and shredded carrots.