Cook the noodles according to the package directions. Drain and rinse under cold water. Set aside.

In a large skillet, heat the vegetable oil over medium-high heat. Add the grated ginger and minced garlic and cook for 1 minute, until fragrant.

Add the soy sauce, rice vinegar, sugar, black pepper, and chili flakes (if using) to the skillet. Stir to combine.

Add the sliced mushrooms, bok choy, green onions, carrots, and red bell pepper to the skillet. Cook for 5-7 minutes, until the vegetables are tender.

Add the cooked noodles to the pot of boiling broth. Ladle the vegetable mixture into the pot and stir to combine.

Serve the soup hot in serving bowls.