Rub the mushroom caps with a damp cloth to clean. Remove stems and reserve for Round 2 recipe "Stuffed Zucchini".
In a small bowl combine the grill seasoning, oil, garlic, Worcestershire sauce and vinegar. Brush over the mushrooms making sure to coat completely. Reserve 2 tablespoons of the marinade for the aioli.
Heat a grill pan or outdoor grill on medium heat. Put the mushrooms, gill side down, onto the grill and grill about 4 to 5 minutes on each side. After you flip the mushrooms top each with 2 tablespoons of Cheddar.
While the mushrooms are grilling, make the aioli sauce by combining the remaining 2 tablespoons of the marinade with the mayonnaise in a small bowl. Spread each hamburger bun with aioli sauce. Remove mushrooms from the grill and serve on the toasted buns with lettuce and tomato slices.