Preheat the oven to 400°F. Brush the cut sides of the bagel with the melted butter and toast for 8 to 10 minutes, until lightly browned. Sprinkle the cheese over the bottom halves of the bagels and bake for another 4 to 5 minutes, until melted.
Meanwhile, place a medium skillet over medium-high heat. Add the bacon and cook for about 6 minutes, until golden brown and crispy on both sides. Drain on paper towels and set aside.
Crack the eggs into the same pan and cook until the whites are set. Place the bacon and fried eggs on the bagels and place the top halves on top.