Prepare the filling by mixing ground beef, shredded Napa cabbage, chopped green onions, minced garlic, grated ginger, soy sauce, sesame oil, vinegar (if using), white pepper, and salt in a bowl. Let sit for 15 minutes.
Assemble the dumplings by placing 1 teaspoon of filling on each gyoza wrapper. Wet edges and seal tightly.
Cook the gyoza in a skillet with oil until golden brown. Add water to steam them until cooked through.
In a pot, combine chicken broth, water, soy sauce, ginger, sesame oil, and black pepper; simmer while adding any optional vegetables.
Add cooked gyoza to the broth and heat through before serving hot.