Place a double layer of paper towels on a plate and add the bacon to a medium-sized sauté pan. Place the pan over medium heat and sauté the bacon until it's crisp, about 7 minutes. Transfer it to the paper towel-lined plate to drain. Crumble the bacon, add it to a small mixing bowl and set aside.
Add the onion and Serrano pepper to the pan with the bacon fat and sauté over medium-low heat, stirring periodically, until the onions are well caramelized, about 10 minutes.
Add the onion mixture to the bowl with the crumbled bacon and season to taste with salt and pepper. (.) Place the 4 bagel halves, round side down on a clean, dry surface and drizzle each one with about ½ tablespoon of the syrup.
Then divide about half of the cheese slices to the bottom 2 bagel halves. Divide the bacon-onion mixture evenly between them and top them with the remaining cheese slices and then the top halves of the bagels.
You can now grill the sandwich on a grill, in a sauté pan, or in a panini press. (If you use a pan or stove-top grill, use high heat, and a flat-bottomed metal spatula to firmly press down on the sandwich, then turn the heat to low and cover the pan with a lid or foil until the cheese has melted, about 4 minutes or so.)
While the sandwich is cooking, use the same pan you cooked the onions in to fry 2 eggs. (If necessary, add a bit of olive oil to the pan and crack the eggs into it. Sauté just until cooked, about 4 minutes.)
Carefully place 1 fried egg on top of each grilled bagel sandwich and sprinkle with a bit of salt and pepper.
Serve!