Cook the Ham: In a skillet over medium heat, melt butter. Add the ham slices and cook until browned on both sides. Set aside.

Fry the Eggs: In the same skillet, add more butter if needed. Crack the eggs into the skillet and cook to your desired doneness—sunny-side up, over-easy, or scrambled. Season with salt and black pepper. Once nearly done, place a slice of cheddar cheese over each egg to melt.

Toast the Ciabatta: While the eggs are cooking, toast the ciabatta rolls in a toaster or skillet until golden and crispy. Mix the mayo and dijon together.

Assemble the Sandwich: Spread Dijonnaise on the bottom halves of the toasted ciabatta. Layer the fried ham and eggs with melted cheddar. Top with the other ciabatta halves and cut in half.