Preheat oven to 375 degrees F (190 degrees C).
Place Red Bell Pepper (2) on a baking sheet, and roast for 30 minutes, until flesh is softened and skin is blackened. Remove red bell peppers from oven, and place in a small glass bowl. Cover bowl with cellophane.
Let red bell 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.
Meanwhile, use a spoon to scrape gills from underside of Portabella Mushroom Cap (8).
In a small bowl, combine Olive Oil (1/4 cup), Worcestershire Sauce (1 tablespoon), Balsamic Vinegar (1 tablespoon), half of the Garlic (1 1/2 clove), Kosher Salt (1/2 teaspoon), and Ground Black Pepper (1/2 teaspoon). Whisk to combine.
For aioli, in a blender or mini-prep food processor, add Fresh Basil Leaf (1 cup), remaining half of the Garlic (1 1/2 clove), and Mayonnaise (3/4 cup). Pulse until basil is incorporated into the sauce. Add Lemon Juice (1 tablespoon), and pulse again.
Scrape aioli into a small bowl (it may look curdled). Add Mayonnaise (1/4 cup), and whisk briskly. It should smooth right out.
Heat grill to medium-high heat, 375-400 degrees F (190-200 degrees C). While grill heats, brush portabella mushroom caps, Radicchio (1), and Red Onion (1) with the marinade.
Grill portabellas, raddicchio, and red onion until flesh is softened, and exterior begins to take on some grill marks, 8-10 minutes. Toast Whole Wheat Hamburger Bun (6) on the grill until lightly browned.
To build burgers, place portabellas on the whole wheat hamburger buns, dress with a sprinkle of Smoked Gouda (1 cup), a few red onion rings, raddicchio, red bell peppers, and Tomato (to taste). Top with basil aioli. Enjoy!