Shape the ground beef into 4 patties and season both sides with salt & pepper, pressing a shallow indent into the center of each one so it stays flatter as it cooks. The patties should look evenly formed but not packed tight, which helps them stay tender instead of firm.
Heat a skillet or grill pan over high heat, then add the patties and cook for 2 to 3 minutes on the first side. You should hear an immediate sizzle, and the edges should start to brown and crisp rather than turn gray.
Flip the patties, add cheese right away, and cook 2 to 3 minutes more depending on how done you like them. The cheese should soften and start to drape over the sides, and covering the pan briefly can help it melt more evenly.
Transfer the patties from the pan and let them rest for a minute. This short rest helps the juices settle back into the meat instead of running into the lettuce wrap.
Cook the bacon and fried egg. If you use the same pan, the egg picks up extra savory flavor from the drippings and the edges can turn lightly crisp.
Place 1 cheeseburger patty in a lettuce wrap, then add ketchup of choice, bacon, fried egg, and green onion. The wrap should feel sturdy enough to hold everything without collapsing, especially if you use crisp romaine or iceberg.
Serve right away. The burger is at its best while the cheese is melty, the bacon is crisp, and the lettuce is still cold and snappy.