Rinse and chop all of the vegetables. Place remaining 3 tablespoons of sesame oil into a large skillet or wok over medium heat. Add onion and garlic and gently sauté just until soft but not brown. Next, add green cabbage and mushrooms, stirring occasionally until soft. Continue with broccoli and red bell pepper until broccoli turns bright green but is still crisp. Add in red cabbage and soy sauce and mix well. Remove from heat.