In a heavy, medium saucepan, heat the vegetable oil over medium-high heat. Add the ginger and cook, stirring, for 1 minute. Add the mushrooms and cook, stirring occasionally, until soft, 2 to 3 minutes. Add the chili paste and brown sugar and cook, stirring often, for 30 seconds. Add 1 cup water, the chicken broth and soy sauce. Stir in the cabbage stem pieces and bring to a boil, then lower the heat and simmer for 25 minutes.
Meanwhile, in a pot of boiling, salted water, cook the pasta according to package directions. Drain and rinse under cold water, then toss with 1 tablespoon sesame oil.
Stir the cabbage leaves, beef and shrimp into the soup and bring to a boil until the shrimp is cooked through. Add the cilantro and remaining 1 tablespoon sesame oil. Divide the noodles among 4 to 6 bowls and add the hot broth with the vegetables, meat and shrimp.