Heat the vegetable oil in a large pot over medium-high heat. Add the lamb pieces and brown them on all sides, about 6 minutes total. Remove the lamb to a plate.
Lower the heat to medium. Add the onion, garlic, and ginger to the pot. Cook for 3 minutes until the onion softens.
Return the lamb to the pot. Stir in the soy sauce and beef broth. Bring the mixture to a simmer.
Add the pearl barley. Cover the pot and cook on low heat for 45 minutes, stirring once or twice, until the barley is tender and the lamb is soft.
Stir in the bok choy halves. Cook uncovered for 5 minutes until the bok choy wilts but still holds its shape.
Ladle the stew into bowls and top with sliced green onions before serving.