Preheat the grill or broiler oven to 325◦F. Rinse visible soil from mushrooms and remove the stems. Baste the mushrooms with olive oil. Place on grill or broiler for approximately 15-20 minutes, turning them over in the middle and checking often to make sure they are not getting charred.
Remove from grill or broiler and place on lettuce leaves, gills up. Divide sauerkraut and scoop into the middle of the mushrooms. Garnish with avocado.