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Lightly grease a 9" x 13" x 2" casserole or baking dish. Set aside. Try to avoid a dish that's too deep, or your casserole won't cook well in the center.
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To make the cheese sauce: Melt the butter in a large saucepan. Add the flour and cook for about 3 to 4 minutes, or until the mixture just begins to color.
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Remove the pan from the heat and whisk in the milk. This will help keep the sauce lump-free. Return to the heat and add the shredded cheese 1/2 cup at a time. Whisk until the cheese melts before adding more cheese. When the mixture is thick and smooth, remove from the heat. Add salt and pepper to taste; set aside.
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To make the filling: Melt the butter in a large skillet. Sauté the ham and your choice of veggies for 5 to 8 minutes, until they begin to brown along the edges.
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Pour the beaten eggs into the hot skillet with the ham and veggies. Scramble until just barely set, then fold in the cheese sauce. Add additional milk if the mixture seems too thick; you're looking for cake-batter consistency.
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Pour the cheese and egg mixture into the prepared pan.
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Cover the egg mixture with an even layer of crumbs, and drizzle with the melted butter. Sprinkle with the paprika, or salt and pepper if you prefer.
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At this point you can chill the casserole from 1 to 12 hours in the fridge, tightly wrapped in plastic wrap.
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When you're ready to bake, remove the pan from the fridge and preheat the oven to 350°F. Bake for 55 to 65 minutes, or until the casserole is bubbling hot in the center. Serve hot.