Bring a pot of water to a boil, salt it, and add noodles. Cook until just done, about 3 minutes. Drain in a colander and run under cold water. Toss noodles with sesame oil to keep them from sticking together, and set aside.
Put peanut oil in a large skillet over medium-high heat. When it's hot, add ginger and cook, stirring, until just fragrant, about 1 minute. Add protein and cook until no longer pink and is starting to brown around the edges.
Add cabbage and carrots to skillet and stir; sprinkle with salt. Continue to cook until vegetables soften, adding a bit of water, as needed to keep them from sticking.
Meanwhile, stir together in a small bowl ketchup, soy sauce, Worcestershire sauce, mirin, and hot sauce. When vegetables are soft and any liquid has evaporated, add noodles and sauce to skillet. Toss to coat everything well and cook until noodles are warmed through. Serve, topped with chopped scallions.