Melt the butter over low heat in a pot on the stove.
Add the flour and whisk together for 2 minutes.
Gradually add the milk, whisking continually, until the sauce thickens.
Remove from the heat and add the salt, pepper, nutmeg and half the cheese. Mix.
Layer the Mushrooms
Place the mushrooms on a clean work surface and tuck a slice of ham into the cavity of each one.
Place 3–4 tablespoons of béchamel sauce on top and sprinkle with the remaining grated cheese.
Arrange on an open grill and cook slowly, well above the coals, until the mushrooms are done and the cheese has melted. (As an alternative: bake for about 20 minutes in the oven at 200°C, then brown quickly under the grill right at the end.)
Serve with sliced baguette and a fresh green salad.