Melt butter in a skillet over medium heat. Fry eggs to your desired doneness, 3 to 6 minutes. Season with sea salt and pepper. Transfer to a plate.
Cook ham in the same skillet until golden brown, about 2 minutes per side.
Place a piece of cinnamon roll dough in the waffle iron; cook until golden brown, about 5 minutes. Repeat with remaining dough to create 10 cinnamon roll waffles. Place a slice of Cheddar cheese on half of the cinnamon roll waffles; add an egg and a slice of ham. Cover each sandwich with another cinnamon roll waffle.
Combine the cream cheese icing from the cinnamon rolls package with orange zest; mix until combined. Drizzle on top of the sandwiches.