In a medium bowl mix together all the patty ingredients with your hands. Gently shape the mixture into four patties of equal size and about 2.5 centimeters thick. With your thumb or the back of a spoon, make a shallow indentation about 2.5 centimeters wide in the center of each patty to prevent it from doming as it cooks. Refrigerate the patties until ready to grill.
In a skillet over medium heat, fry the bacon until crisp, 10 to 12 minutes, turning occasionally. Transfer the bacon to paper towels to drain. Pour off all but 2 tablespoons of the bacon fat from the skillet and reserve for another use or discard.
Prepare the grill for direct cooking over medium-high heat (200º to 260ºC).
Brush the cooking grates clean. Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness (70ºC), 9 to 11 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, place a slice of cheese on each patty to melt, and toast the rolls, cut side down, over direct heat. Remove from the grill.
Brush the cooking grates clean. Grill the patties over medium-high heat, with the lid closed, until cooked to medium doneness (70ºC), 9 to 11 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, place a slice of cheese on each patty to melt, and toast the rolls, cut side down, over direct heat. Remove from the grill.
Return the skillet to the stove over medium heat and warm the bacon fat. Crack the eggs into the skillet, turn down the heat to medium-low, and cook until the whites are just firm and opaque and the yolks have begun to thicken, about 2 minutes. Carefully flip the eggs over and continue to cook until the yolks are cooked but still slightly runny, 30 seconds to 1 minute more. Season with salt and pepper.
Build a burger on each bun with two tomato slices, two bacon slices, a patty, and an egg. Serve warm.
