Place carrots and turnips in the bottom and up the sides of a 4-quart or larger slow cooker. Top with pork and scallion whites. Bring broth, water, soy sauce, sherry, brown sugar, ginger, vinegar, chile-garlic sauce to taste and garlic to a simmer in a medium saucepan over medium-high heat. Pour over the pork and vegetables. Nestle star anise pod (or aniseed) and cinnamon stick into the stew. Cover and cook until the pork and vegetables are tender, 3 to 3 1/2 hours on high or 5 1/2 to 6 hours on low.

Discard the star anise pod and cinnamon stick. Skim or blot any visible fat from the surface of the stew. Add the cornstarch mixture, cover and cook on high, stirring 2 or 3 times, until slightly thickened, 10 to 15 minutes. Serve sprinkled with scallion greens and sesame seeds.