1. In a skillet, cook the bacon over medium heat until crisp. Transfer to paper towels to drain.
2. Set the Monterey Jack slices on 1 piece bread, and toast both slices of bread under the broiler until lightly browned and bubbly. Spread the slice of bread without cheese with mayonnaise, and then top with bacon, then lettuce, then tomato.
3. Melt butter in a small nonstick pan. Add the egg and fry over moderate heat, turning once, until crisp around the edge, about 4 minutes; the yolk should still be runny. Slide the egg onto the lettuce; close the sandwich and eat immediately.