Special equipment: an 8-inch square baking dish; a meat grinder with 1/2-inch grinder plate attachment
Preheat the oven to 400 degrees F. Butter the baking dish.
Put 3 tablespoons butter and the flour in a medium saucepan and cook, stirring, until the mixture begins to brown and thicken, about 3 minutes. Remove from the heat and whisk in the milk. Bring the mixture to a simmer over high heat and cook, stirring, until thickened, 5 to 6 minutes. Set aside.
Put the egg yolks in a large bowl. Using a 1/2-inch grinder plate attachment, feed the ham through the grinder, allowing it to fall into the bowl with the yolks. Stir in the thickened milk mixture and all but 1 tablespoon of the breadcrumbs until combined.
Beat the egg whites to stiff peaks using an electric hand mixer, and then fold them into the ham mixture. Transfer the mixture to the baking dish. Sprinkle the top with the remaining tablespoon of breadcrumbs. Bake until firm and golden brown, about 30 minutes. Serve hot.