Fill a large heavy-bottomed pot or Dutch oven halfway with water and bring to a boil over high heat.
Add the brisket pieces and return the water to a boil. Immediately drain the meat through a colander and rinse any scum off of it with cold water; set aside.Wash and dry the pot.
Heat the oil in the pot over medium-high heat until shimmering.
Add the ginger and garlic and cook, stirring occasionally, until fragrant and starting to brown, about 1 minute. Return the meat to the pot, add the chu hou paste, and stir to coat.
Add the chicken broth, measured water, star anise, sugar, and measured salt and stir to combine.Bring the mixture to a boil. Cover with a tightfitting lid, reduce the heat to low, and simmer until the meat is tender, about 2 1/2 to 3 hours.Meanwhile, peel and cut the turnips or daikon into 1-1/2-inch cubes, place in a large bowl, and cover with a damp paper towel or plastic wrap. Trim and cut the scallions into 1-inch pieces, place in a small bowl, and cover with a damp paper towel or plastic wrap. Refrigerate the vegetables until the meat is tender.
Add the turnips or daikon to the pot and stir to combine. Cover and simmer, stirring halfway through the cooking time, until tender, about 30 to 40 minutes.
Remove the pot from the heat, stir in the scallions, and let sit uncovered for 5 minutes to allow the scallions to soften slightly.
Remove and discard the star anise pods. Taste and season with salt as needed.