Heat the oil in a large saucepan or Dutch oven over medium heat. Add the onion, garlic, and ginger and cook, stirring, until fragrant, 1 to 2 minutes.
Add the broth. Stir in the parsnips, carrots, celery, turnip, and ½ teaspoons salt. Bring to a boil. Reduce heat and simmer until the vegetables are tender, 15 to 20 minutes.
Meanwhile, using a fork or your fingers, shred the chicken meat, discarding the skin and bones.
Add the chicken, peas, and scallions to the saucepan and cook until heated through, 3 to 4 minutes. Serve with the biscuits, if using.