How to Make It
Heat a medium saucepan over medium-high heat. Add 1 tablespoon sesame oil to pan; swirl to coat. Thinly slice mushroom caps. Add mushrooms and carrot to pan; sauté 5 minutes or until tender. Add peppers, garlic, and ginger; cook 30 seconds, stirring constantly. Add stock; bring to a boil, scraping pan to loosen browned bits. Cover and simmer 10 minutes. Remove from heat; stir in soy sauce and honey. Keep warm.
Heat a wok or large skillet over high heat. Add remaining 1 tablespoon sesame oil to pan; swirl to coat. Add broccoli; stir-fry 3 minutes. Add bok choy; stir-fry 1 minute.
Place noodles in a colander; rinse under hot water to separate noodles. Drain; divide noodles evenly among 4 bowls. Top evenly with broccoli, bok choy, and stock mixture.
Heat a large nonstick skillet over medium heat. Add canola oil to pan; swirl to coat. Crack eggs into pan; cook 4 minutes or until whites are just set. Top each bowl with 1 egg. Drizzle servings evenly with rice vinegar.