1. Prepare a grill to medium high heat for direct heat grilling.
2. Toss the mushroom caps in a bowl with 2 tablespoons oil, 1 teaspoon salt and 3/4 teaspoon pepper. Place the mushrooms, gill-side up on the grill and cook, covered, turning over occasionally until soft, 14 to 16 minutes. Transfer the mushrooms to several layers of paper towels and press out any remaining liquid. Let cool.
3. Stir together 1/3 cup mayonnaise with chili sauce in small bowl, and set aside.
3. In separate bowl, combine the remaining 1/3 cup mayonnaise, Cheddar, and Muenster cheeses. Place 4 mushroom caps, gill-side up on a work surface. Mound a quarter of the cheese mixture on each mushroom cap. Top with the remaining mushrooms, pressing together firmly.
4. Beat the eggs in a shallow bowl. Put the panko in another bowl. Dip each stuffed mushroom patty in the egg, turning to coat, then dredge in the panko to coat completely.
5. Heat about 3-inches oil in a medium saucepan until a deep-fry thermometer registers 375 degrees F. Fry the mushroom burgers, in batches, if necessary, until golden brown, 2 to 3 minutes. Drain on paper towels. Season with salt.
5. Tuck the 'shroom burgers into the rolls and top with lettuce, onion, and the reserved chili sauce. Serve.