Bring a large pot of water to a boil over high. Add noodles, and cook 1 minute less than package directions. Drain and rinse with cold water until completely cool; drain well. Set aside.
Heat 2 tablespoons vegetable oil in a large skillet over high until shimmering. Stir in beef and salt; cook, stirring often, until no longer pink, about 1 minute.
Transfer beef to a plate; set aside. Add remaining 1 tablespoon oil to skillet; stir in cabbage, mushrooms, onion, scallions, and carrot. Cook, stirring often, until vegetables are crisp-tender, 3 to 4 minutes. Stir in noodles and beef; cook until heated through, about 1 minute.
Stir tonkatsu sauce, ketchup, and soy sauce into noodle mixture; cook, stirring often, until sauce thickens, 2 to 3 minutes. Remove from heat; divide noodle mixture evenly among 4 plates, and serve.