Combine beef stock, hoisin sauce, sherry wine, and soy sauce in a bowl.

Set the sauce mixture aside.

In a large pan or Dutch oven, heat oil over medium-high heat.

Brown the lamb shanks on all sides in the hot oil.

Add minced garlic, minced ginger, diced carrots, diced onions, and chopped green onion to the pan.

Cook until the vegetables soften slightly, about 5 minutes.

Pour the sauce mixture over the lamb shanks and vegetables.

Bring the mixture to a boil.

Reduce the heat to low, cover the pan, and simmer for approximately 2 hours, or until the lamb is very tender.

Stir occasionally to ensure even cooking.

Remove the lamb shanks from the pan.

Skim off any excess oil from the surface of the sauce.

Use a hand blender or immersion blender to puree the vegetables and juices until smooth.

If a thicker consistency is desired, dissolve cornstarch in a small amount of water and add it to the sauce.

Simmer for a few minutes until the sauce thickens.