1. Broil or grill mushrooms on both sides until just tender, about 5 minutes per side. Sprinkle with salt and pepper.
2. Cook onions in olive oil over medium heat until very tender, stirring occasionally, about 10 to 12 minutes.
3. Cut avocado in half, remove the pit, and peel and cut into slices.
4. Toast buns. Top bun bottom with grilled portabella, 1/4 of the onion and avocado slices. Top with bun top.