Heat oven to 375 degrees. Place bell peppers on a baking sheet and roast for 30 minutes until flesh is softened and skin is blackened. Remove peppers from oven and place in a small glass bowl. Cover bowl tightly with cellophane. Let peppers cool until they are comfortable to handle. Peel the skins from the peppers. Pull peppers apart into large chunks and discard seeds and stems. Set aside.
While peppers cook, use a spoon to scrape the gills from the underside of the mushroom caps. Set aside.
In a small bowl combine olive oil, worstershire, balsamic, garlic, salt and pepper. Whisk to combine.
For aioli: In a blender or mini-prep food processor add basil leaves, garlic and 3/4 cup of mayonnaise. Pulse until basil is incorporated into the sauce. Add a lemon and pulse again (note: I have a Vitamix blender which can turn concrete into a frothy drink - which is why the color of my aioli is green. With a regular blender/food processor, it probably won't achieve the same color, but the flavor will be there. Promise.) Scrape aioli into a small bowl -- it may look curdled. Add the remainder of the mayonnaise and whisk briskly. The aioli should smooth right out.
Heat a grill to a medium high heat - about 375-400 degrees. While grill heats brush mushroom caps and onion slices with marinade.
Grill mushroom caps and onion until flesh is softened and exterior begins to take on some grill marks. About 8-10 minutes. Toast six buns on the grill for just a few minutes, until lightly browned.
Build burgers. Spread basil aioli on the top and bottom buns. Place mushrooms on the buns, dress with a sprinkle of cheese, a few onion rings and roasted red pepper and tomato.