Warm the vegetable oil in a large pot over medium heat.
Sauté the garlic and ginger until fragrant, about 30 seconds.
Add mushrooms and carrots; cook, stirring, until slightly softened, 3 to 4 minutes.
Pour in the chicken broth. Stir in soy sauce and rice vinegar, then bring to a gentle simmer.
Slide in the frozen gyoza. Simmer gently until cooked through and floating, 5 to 6 minutes.
Stir in the bok choy and cook until tender-crisp, about 2 minutes.
Finish with sesame oil, salt, and white pepper. Taste and adjust seasoning.
Ladle into bowls. Top with scallions and a drizzle of chili oil, if using.