In a large skillet, melt 1 tablespoon of butter over medium heat. Add the slices of ham and cook for about 2-3 minutes on each side until heated through and slightly crispy. Remove from the skillet and set aside.
In the same skillet, crack the eggs and cook to your desired doneness (sunny-side up, over-easy, or scrambled). Season with salt and pepper.
While the eggs are cooking, place a slice of cheddar cheese on each piece of ham in the skillet for the last minute of cooking to melt the cheese.
Spread the remaining tablespoon of butter on the toasted English muffins.
Assemble the sandwiches by placing a slice of ham with melted cheese on the bottom half of each muffin, followed by the cooked egg, and then the top half of the muffin.
If desired, add hot sauce or your favorite condiment before serving.