Put a cast-iron skillet over medium-low heat. Butter the cut sides of the bagels. Lay the bagel halves in the skillet butter-side down and toast until golden, 2 to 3 minutes. Set aside.
In the same skillet, throw in the sliced onions and 1 tablespoon of the butter and cook slowly, stirring occasionally, until the onions are soft and golden, about 20 minutes. Remove to a plate.
Meanwhile, in a small bowl, mix together the mayonnaise, mustard, some hot sauce and 1 teaspoon each salt and pepper. Set aside.
Mix together the hot and maple sausage in a medium bowl. Form the mixture into six 1/2-inch-thick patties, slightly bigger than the diameter of the bagels (since the burgers will shrink during cooking).
Put the same skillet over medium heat, add the sausage patties and fry on both sides until cooked through, 5 to 7 minutes. Top each patty with 2 slices of Cheddar and cover the skillet with a lid to facilitate melting, 1 to 2 minutes.
In a separate nonstick skillet over medium-low heat, fry the eggs in 1 tablespoon of butter and sprinkle with salt and pepper. Cook until set on one side, then flip and cook the second side to desired doneness.
Build the sandwiches by spreading the mayo/mustard mix on the bottom half of each bagel, then top with a sausage patty, some caramelized onions and a fried egg. Spread more of the mayo/mustard mix on the top half of the bagels and place on top of the eggs to finish. Serve warm or wrap in foil to transport.