For the sauce, melt the butter in a pan over medium heat. Add flour, then beat hard until you have a paste. Take off heat; let cool 2 minutes, then gradually add milk, whisking constantly. Return to heat, and add mustard and nutmeg; simmer gently for 10 minutes, whisking frequently so the sauce doesn't burn. Once the sauce thickens to the consistency of tomato sauce, take it off the heat. Add the cheese (keep a little to garnish). Season to taste.
Preheat oven to 350 degrees. Flatten the bread with a rolling pin; brush both sides with melted butter. Line 6-cup muffin tin with the bread slices, pressing them with the bottom of a glass.
Divide the ham among the muffin cups. Crack an egg into each (if the egg seems too big, pour a little of the white away before using). Top with 2 tablespoons sauce, then sprinkle with cheese and pepper. Bake 15-20 minutes, and serve.