In a Japanese donabe or a wide, low pot, combine the chicken broth, garlic, and remaining 2 tsp. ginger over medium-high heat. Bring broth to a boil, then reduce to a simmer. Whisk in the miso and remaining 1 tsp. soy sauce. Taste and add salt if needed. With wet hands, form tablespoon-size meatballs and add to the broth. Add the cabbage, tofu, carrots, and mushrooms. Cover and return to a gentle simmer; cook for 5 minutes.