Slice each bun in half, and place the slices on a work surface. To assemble a sandwich, place the toppings in this order: bottom of cinnamon bun, 1 slice Cheddar cheese, 1 slice of Brie cheese, apple slices, shredded ham, 1 slice of Brie cheese, 1 slice of Cheddar cheese, top of cinnamon bun. Repeat for 2nd sandwich, and gently press each sandwich together.
Melt butter in a skillet over medium heat.
Whisk eggs and milk together in a bowl.
Dip both sides of each sandwich quickly into the egg mixture, and pan-fry the sandwiches until golden brown on each side, about 2 minutes per side.
