Preheat the broiler. In a large skillet, cook the bacon over medium-high heat until the fat has rendered and the meat is partially cooked, 2 to 3 minutes. Transfer the bacon to a foil-lined baking sheet and sprinkle with the brown sugar; reserve the fat. Broil the bacon until the sugar melts, about 1 minute. Transfer to a paper-towel-lined plate, leaving the broiler on.
In a large bowl, combine the onion, vinegar and olive oil. Whisk in 2 tablespoons of the reserved bacon fat. Season with salt and pepper. Add the romaine and toss. Arrange the salad on 4 plates and crumble the reserved bacon on top.
Place the bread on a baking sheet and broil until toasted on one side. Flip, then top each slice with the cheese. Broil until the cheese is browned and bubbling. Cut into narrow wedges and tent with foil to keep warm.
Heat the skillet with the remaining bacon fat over medium-high heat. Fry the eggs until the whites are opaque and the yolks are still runny, about 3 minutes. Place 1 egg sunnyside up on each portion of salad. Serve with the cheese toasts.