- Melt the 1 Tbsp. butter in a small pot set over medium heat. Mix in the flour and cook 2 minutes. Very slowly whisk in the milk. Bring to a simmer and whisk until you have a lightly thickened white sauce; season with salt, pepper and nutmeg. Cover sauce, remove from heat and set aside until needed.
- Butter one side of each bread slice. Set two bread slices on a work surface, buttered side down. Spread the tops of those slices with the mustard. Now divide and top the mustard on each bread slice with ham, tomato, avocado and two thirds of the grated cheese. On top of this, set the other two bread slices, buttered side up.
- Set a large skillet over medium heat. When hot, add sandwiches and cook until golden brown, about three minutes per side. Transfer the sandwiches to a non- stick baking sheet.
- Set an oven rack 6” below your oven’s broiler. Preheat the broiler. Spoon and spread white sauce over each sandwich (it’s OK if it spills over the sides).
- Top each sandwich with some of the remaining grated cheese. Broil the sandwiches until cheese and the sauce begin to bubble and brown, about 2 minutes. Set a sandwich on each of two plates, scooping up and including the broiled white sauce that dripped onto the pan. Sprinkle the croque monsieur with chopped parsley, if using, and serve.
Recipe Options: Instead of Gruyere or Swiss, try another type of tangy cheese, such as Gouda or Jarlsberg.