Bring a large saucepan filled with salted water to a boil and grate garlic cloves and ginger.
Heat vegetable oil over medium heat in a large pot, then sauté grated garlic and ginger for 30 seconds.
Add chicken broth and coconut milk to the pot, then stir in soy sauce, fish sauce, sesame oil, brown sugar, Thai red curry paste, and lime juice until the curry paste dissolves.
Bring the soup to a boil, then reduce heat and simmer for 5 minutes.
Meanwhile, cook the gyozas in the boiling water according to package instructions, about 2 to 3 minutes, then drain.
Divide cooked gyozas between bowls, ladle hot coconut curry soup over them, and garnish with chopped chives and optional vegetables.