Heat 2 teaspoons oil in a large nonstick skillet over medium heat. Add mushrooms, onion, and bell pepper; cook 10 minutes or until liquid evaporates, stirring occasionally. Add garlic; cook 1 minute. Stir in thyme, salt, and black pepper. Remove 1 cup mushroom mixture, and set aside. Increase heat to medium-high. Add broth and cornstarch to pan; bring to a boil. Cook 1 minute, stirring constantly. Keep warm.
Combine reserved 1 cup mushroom mixture, brown rice, breadcrumbs, and eggs; mix well.
Heat remaining 1 tablespoon oil in a nonstick griddle or large nonstick skillet over medium-high heat. Spoon about 1/2 cup rice batter per pancake onto hot pan, spreading each to a 4-inch diameter. Cook 5 minutes on each side or until bottoms are golden brown. Serve with mushroom sauce.