Combine ingredients for garlic Herb mayonnaise; set aside. In a medium bowl, combine rice, eggs, bread crumbs, cheese, oregano, salt and pepper. In a non-stick skillet over medium-high heat, heat 1 tsp (5 mL) oil. Saute onion and carrot 2 minutes. Add mushrooms and garlic and saute 1 to 2 minutes longer. Add cooked vegetables to egg mixture; mix thoroughly. Shape mixture into 4 patties.
In same skillet over medium high heat, heat the remaining 2 tsp (10 mL) oil. Saute burgers 2 to 3 minutes per side or until lightly browned. Spread garlic herb mayonnaise on both halves of rolls. Place lettuce on bottom half, top with veggie burgers and garnish to taste.