Pat the lamb shanks dry and season lightly with salt. Heat the vegetable oil in a large Dutch oven over medium-high heat. Brown the shanks on all sides, then remove them to a plate.
Lower the heat to medium. Add the onion, garlic, and ginger to the same pot. Cook until the onion softens and the mixture smells fragrant.
Stir in the shiitake mushrooms and cook for two minutes until they start to soften.
Pour in the soy sauce, rice vinegar, and beef broth. Add the honey and star anise pods. Return the lamb shanks to the pot and bring the liquid to a simmer.
Cover the pot and reduce the heat to low. Braise for two and a half to three hours, turning the shanks once, until the meat is very tender.
Remove the star anise pods. Serve the shanks and mushrooms over steamed rice, topped with the sliced green onions.