Heat 2 tablespoons vegetable oil at the bottom of a soup pot over medium-high to high heat and add protein of choice and stir fry: shrimp until just pink, chicken or pork until lightly browned. Add mushrooms, carrots and garlic, cook for 2 minutes, then add cabbage and wilt it, 5 minutes.
While cabbage cooks, add 2 tablespoons vegetable oil to a skillet and arrange pot stickers in it. Add 2/3 cup water and turn on to medium-high heat. Cook for 8 minutes to evaporate the water, then let the pot stickers brown 1 minute more.
Add scallions, soy sauce and black pepper and stir to combine, then add stock and bring to a boil. Stir in beaten eggs and continue to stir in pot until they are cooked and scattered throughout the stoup.
Place a few pot stickers in the bottom of a soup bowl and top with soup.