Place grill bag on hot grill, window side up, and cook until mushrooms are softened, 12-15 minutes. Do not close grill; heat from a closed grill will cause grill bag window to melt.
After 5 minutes, place patties on hot grill. Grill until patties reach a minimum internal temperature of 165 degrees, 3-4 minutes per side.
If using 20 oz ground beef, cook four patties following the same instructions. If using ground pork, follow same instructions, cooking until turkey reaches a minimum internal temperature of 160 degrees.
Remove patties from grill and top with cheese.
Place buns on grill, cut-side down, and toast, 30-60 seconds.