In a small bowl, whisk together the mayonnaise, mustard and horseradish, to taste. Set aside.
Place the top of 1 muffin on the ham steak and cut around to make a circle. Repeat for 2 ham slices. *Note, more circles can be cut for additional servings.
Clean, oil and preheat the grill to medium-high heat. (This can also be done in the house on the burner). Place the ham on the grill and cook for about 4 to 5 minutes per side or until lightly charred.
Remove the ham from the grill and set on the warm side of the grill.
Place a cast iron skillet over medium-high heat.
Grease the inside of a mason jar lid ring or an open tin can, (ie tuna fish can). Place the can in the cast iron skillet and crack 2 eggs in the middle. Lightly scramble the eggs with a fork. Season with salt and pepper, to taste.
Cook for about 2 minutes or until the eggs set up a bit. Remove the can and flip the eggs over. Continue cooking until the eggs have cooked through, about 2 more minutes. Repeat with remaining eggs.
Place the eggs on the warm side of the grill and top with 1 slice of cheese. Top the ham slices with 1 slice of cheese. Let set a minute or so, or until the cheese begins to melt.
Meanwhile, butter the undersides of the English muffins and place on the grill to toast.
Spread the mayonnaise mixture on both sides of the muffins. Add the ham, eggs and top slice of the muffin. Serve immediately.