Preheat broiler; place rack in upper middle position.
Heat 1 tablespoon oil in a 10-inch oven-safe skillet over medium-high. Add carrot and white parts of green onions; sauté 3 minutes. Add slaw, ginger, and garlic; cook 3 minutes, stirring occasionally. Add vinegar; cook 1 minute or until liquid evaporates.
Combine 1 1/2 teaspoons tamari, 1/4 teaspoon pepper, and eggs in a bowl, stirring with a whisk. Add egg mixture to slaw mixture in pan, tilting pan to spread mixture evenly. Cook 3 to 4 minutes or until egg is set on the bottom. Place pan in oven; broil 2 minutes or until top is set and golden. Cut into quarters.
Combine stock and flour in a bowl, stirring with a whisk. Heat remaining 1 tablespoon oil in a skillet over medium-high. Add mushrooms; cook 5 minutes or until browned. Add flour mixture; bring to a boil, and cook 1 minute or until mixture is slightly thickened. Stir in remaining 1 tablespoon tamari and 1/4 teaspoon pepper.
Heat rice according to package directions; divide evenly among 4 plates. Top each serving with 1 wedge of the pancake and 1/4 cup mushroom mixture; top evenly with green parts of onions, remaining 1/4 teaspoon pepper, and sesame seeds.