For the honey sauce: Melt the butter in a small saucepan over medium heat. Add the honey and hot sauce and stir to combine. Cook until the sauce comes together and the butter is melted. (Once the eggs are cooked, add the chives and stir to combine.)
For the ham and cheese: Next, add the ham to a large nonstick grill pan over medium heat. Cook the ham for 20 to 30 seconds, then flip and repeat, cooking for 20 to 30 more seconds. Create 2 stacks of 3 slices of ham on the grill pan and top each with 1 slice of cheese; cook until the cheese melts.
For the eggs: Melt the butter in a medium nonstick skillet over medium heat. Individually crack the eggs into a small glass bowl, then add the eggs to the skillet one at a time, leaving space between them; sprinkle with salt and pepper. Cook until the egg whites are just starting to set, about 2 minutes, then continue cooking until they are over easy, about 1 more minute.
To serve: In the same pan as the ham, lightly toast the brioche bun. Spread the honey sauce on the bottom half of the bun and top with the sliced apples. Put the ham and cheese stacks on the apples. Remove the eggs from the pan with a large spatula and gently place on the ham; top with more sauce and the bun lid; serve immediately.