Heat peanut oil in a large nonstick skillet over medium-low heat. Cook garlic and ginger until fragrant, about 1 minute. Add vegetable broth and water. Cover and cook until garlic and ginger are very soft, about 7 minutes. Stir in bell pepper and carrots and cook for 1 minute. Add bok choy and toss until well coated with liquid. Season with sugar and salt and cook until tender, about 4 minutes more. Drizzle with sesame oil.