Pour oil into an 8- to 10-inch nonstick frying pan over high heat; when hot, add mushrooms and stir until lightly browned, about 4 minutes. Stir in the green onions and plum sauce and remove from heat.
In a bowl, gently mix ground beef with soy sauce, sesame oil, garlic, Chinese five spice, and ginger. Shape beef mixture into four equal patties, each about 1/2 inch thick.
Lay patties on a lightly oiled barbecue grill over a solid bed of medium-hot coals or medium-high heat on a gas grill (you can hold your hand at grill level only 3 to 4 seconds); close lid on gas grill. Cook patties until no longer pink in the center (cut to test) or a thermometer inserted in center reaches 160°, 6 to 8 minutes total, turning once to brown evenly. After 5 minutes, lay bun halves, cut side down, on grill and toast lightly, 1 to 2 minutes.
Lay a lettuce leaf on each toasted bun base, top with a beef patty, and spoon mushroom mixture equally over burgers. Add salt to taste and cover with bun tops.