Preheat the oven to 350˚F. Grease an oven-safe 9-inch-square baking dish with butter and set aside.
For the béchamel sauce: Melt the butter in a medium pot over medium-high heat. Once melted, add the flour and stir with a wooden spoon. Allow the mixture to foam and bubble for 2 minutes, stirring occasionally to ensure the contents do not burn. In a steady stream, whisk in the milk until smooth and thickened. Turn the heat to low and gently simmer for 2-3 minutes, stirring occasionally with a wooden spoon. Add the mustard, salt and cheese and stir until the cheese has melted and the sauce is smooth. Remove from the heat.
To assemble the Croque Madame bake: Arrange four slices of bread so that they cover the base of the greased baking dish in one even layer. Pour half of the béchamel sauce evenly over the bread. Layer the ham slices on top of the sauce followed by half of the grated cheese. Top with the remaining four slices of bread. Spread the remaining béchamel sauce over top and sprinkle with the remaining cheese. Bake in the oven for 15 minutes, or until golden brown on top and bubbling around the edges. If the top has not browned, you can finish it under the broiler, set to medium-high, for 2 – 5 minutes, or until golden.
Top with four sunny-side-up fried eggs and cut into squares and serve warm, family-style.
Watch me make a croque monsieur — and then turn it into a croque madame — in the video below.