For the soup
Heat the vegetable oil in a soup pot over medium-high heat. Add the ginger and green onions and stir-fry until aromatic, about 30 seconds. Add the pork and stir-fry until it is cooked through, 4 to 5 minutes. Add the mushrooms and cabbage and stir-fry until the cabbage is tender, 3 to 4 minutes. Add the stock, soy sauce, vinegar, salt, and pepper and bring the soup to a boil. Mix the cornstarch and water together. Slowly add the slurry to the boiling soup, stirring constantly. Slowly stir the eggs into the soup. Hold hot, but do not boil. Add the sesame oil. Serve in heated bowls or cups and garnish each serving with green onions.