Spread half of the cheese on 2 slices of bread and cover each with 1 slice of cooked ham. Cut or hammer this in such a way that it does not protrude. Scatter the rest of the cheese on top except for 2 tablespoons and cover with the other two slices of bread.
For the sauce, melt the butter in the oil in a saucepan over medium heat. Sprinkle in the flour and sweat for 1 minute. Take the pot off the stove, pour in the milk while stirring and put the pot back on the stove. Then stir for 1 minute until a smooth, thick sauce is formed. Take this off the heat for good, add the rest of the cheese and a little pepper, put the sauce aside and keep it warm.
Whisk the egg in a deep plate. Add the prepared croques one after the other and let the egg mixture soak in under pressure.
Preheat the grill and line a baking sheet with aluminum foil. Melt the butter in a pan over medium heat and fry the croques until golden brown on both sides. Place the croques on the prepared baking sheet and cover with the hot sauce. Then slide it under the grill (approx. 10 cm from the grill snake) and grill until golden brown for 4 minutes.
For Croque Madame, a fried egg is added to the filling in addition to ham and cheese.
If you like it hearty, you can coat the bread with Dijon or grainy mustard before filling.