Heat oven to 375°F and place a rack in the bottom third. Generously butter a 6-cup souffle dish and sprinkle with the breadcrumbs, turning to coat evenly and discarding any excess. Place a baking sheet in the oven to heat.
Melt butter in a small pan over medium heat. When it foams, add flour and whisk until the raw cereal taste has cooked out, 3-4 minutes. Gradually add milk, whisking constantly. Bring to a boil, stirring often. Remove from heat and whisk in salt, pepper and nutmeg.
In a large bowl, whisk egg yolks until smooth. Add hot cream sauce, a spoonful at a time, whisking constantly, until about 1/2 cup of sauce has been added. Whisk egg yolks back into sauce until smooth. Add cheese and whisk until melted.
In a clean bowl (or the work bowl of a stand mixer), beat egg whites (on medium if using electric beater) until soft peaks form, about 5 minutes. Increase to high speed and beat until peaks are stiff and glossy but not dry, about 2 minutes.
Fold 1/3 of egg whites into cheese sauce until no streaks of white are visible. Add chives and remaining egg whites and fold gently to combine. Pour into prepared souffle dish.
Place dish on hot baking sheet. Bake until well-puffed, golden brown, and the center is set (it does not move when gently touched), 35-40 minutes. Serve immediately.