Special equipment: a 2-quart (8-cups) souffle dish
Arrange an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Butter the bottom and sides of a 2-quart souffle dish. Set aside.
In a medium saucepan, heat 1/2 stick of butter over medium-low heat. Add the flour and cook, stirring constantly, for 2 minutes. Slowly whisk in the milk until the mixture is smooth and creamy. Bring the mixture to a simmer and cook, stirring constantly, until the mixture coats the back of a spoon, about 8 to 10 minutes. Stir in the nutmeg and season with salt and pepper, to taste. Remove the pan from the heat and add the cheeses, the spinach, chicken, egg yolks, and bread cubes. Stir until combined (mixture will be thick).
In a large bowl, using an electric mixer, beat the egg whites at high speed until they hold stiff peaks, about 2 minutes. Stir 1/4 of the egg whites into the chicken mixture. Using a spatula, fold in the remaining egg whites. Spoon the batter into the prepared dish and bake until the souffle is puffed and the top is firm and golden, about 55 to 60 minutes. Remove from the oven and serve immediately.