Cook the Ham: In a skillet, heat a small amount of butter over medium heat. Add the slices of ham and brown well. Set aside.
Fry the Eggs: In the same skillet, add a little more butter if needed and crack the eggs into the skillet. Cook the eggs as preferred: sunny side up, over easy, or scrambled. Season with salt and black pepper. Melt the cheese over top.
Toast the Ciabatta: While the eggs are frying, toast the ciabatta rolls in a toaster or in the skillet until golden brown and crispy.
Assemble the Sandwich: Spread the Dijon and mayonnaise inside the toasted ciabatta roll. Place fried ham and eggs on the ciabatta. Close the sandwich with the top half of the ciabatta. Cut in half.