Heat vegetable oil in a large pot over medium heat.
Add ground pork or turkey and cook until browned, breaking it up as it cooks, about 5 to 7 minutes.
Add diced onion, minced garlic, and grated ginger to the pot; cook until onions soften and become fragrant, about 2 to 3 minutes.
Pour in the chicken or vegetable broth and bring the mixture to a gentle simmer.
Stir in the coleslaw mix and cook until the vegetables have softened, around 5 minutes.
Whisk together soy sauce, sesame oil, rice vinegar, and black pepper in a small bowl, then add to the pot and mix well.
Allow the soup to simmer for another 5 to 10 minutes so the flavors can develop.
If using, fold in the cooked noodles just before serving and warm through.
Serve the soup hot with a sprinkle of sliced green onions on top.