Heat 1 tablespoon of fat in a saucepan. Stir in flour and sweat briefly. Pour in milk, bring to the boil and stir in broth. Let simmer for about 5 minutes. Season to taste with salt and pepper.
Cover 4 slices of bread with 1 slice of ham and cheese each. Cover with remaining slices of bread. Heat 2 tablespoons of fat in portions in a coated frying pan. Fry the croquettes in it in portions on each side for about 3 minutes and keep warm.
Heat 1 tablespoon of fat in a second pan. Fry the eggs in it until they are fried. Season with salt and pepper. Arrange croques. Spread sauce and eggs on top.