In a skillet, cook the bacon over moderate heat, turning, until crisp, about 8 minutes. Transfer to paper towels to drain.
Set the Monterey Jack slices on 1 piece of toast. Spread the mayonnaise on the other slice of toast, then top with the bacon, tomato and lettuce.
In a small, nonstick skillet, melt the butter. 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 right away.