Whisk mayonnaise, mustard, 2 tsp rosemary and garlic in small bowl to blend, set aside.
Soak eggplant slices in lightly salted water for about 15 minutes Rinse and pat dry. Brush both sides with olive oil and lightly salt and pepper.
Spray grill rack and prepare at medium heat. Brush mushrooms with olive oil. Sprinkle stem side with minced fresh rosemary, thyme and parsley evenly divided between the mushroom caps, top with pat of butter salt and pepper and 2 tsp balsamic vinegar per mushroom. Grill mushrooms until tender (about 10 - 12 mins, do not turn!) Lay slices of eggplant on grill and cook about 8 - 10 mins until tender, turning a couple of times.
Spread Mayo mixture on toasted buns. Place lettuce, eggplant and mushroom on bun bottoms, cover with tops, serve immediately and enjoy!