Egg Noodle Stir-Fry with Broccoli

Red chili pepper gives the dish a bright color and rich fruit flavor. If you can't find it, substitute 1/4 cup thinly sliced Fresno pepper (which is much softer). Choose very young ginger; it will have a thinner skin and less fibrous inner part.
Preheat a medium saucepan over medium-high heat. Add 1 tablespoon sesame oil to the skillet; stir to coat. Thinly slice the mushroom caps. Add the mushrooms and carrots to the skillet; simmer for 5 minutes or until tender. Add pepper, garlic and ginger; cook for 30 seconds, stirring constantly. Add the broth; bring to a boil, scraping off the pan to avoid browning the pieces. Cover and simmer for 10 minutes. Remove from heat; add soy sauce and honey. Keep warm.Preheat a wok or large skillet over high heat. Add the remaining 1 tablespoon of sesame oil to the pan; stir to coat. Add broccoli; stir-fry for 3 minutes. Add the bok choy; stir-fry for 1 minute.Drain the noodles in a colander; rinse under hot water to separate the noodles. Drain; spread noodles evenly over 4 bowls. Sprinkle evenly with broccoli, bok choy, and stock mixture.Preheat a large nonstick skillet over medium-high heat. Add the canola oil to the pan; stir to coat. Break the eggs into a skillet; cook for 4 minutes or until the whites are set. Place 1 egg in each bowl. Drizzle the rice vinegar evenly over the portions.