Yakisoba

2 (3.5-ounce) packages dried ramen noodles or 2 (4-ounce) packages frozen ramen noodles
3 tablespoons vegetable oil, divided
8 ounces thinly sliced rib eye steak
¼ teaspoon kosher salt
4 cups cored and thinly sliced napa cabbage
2 cups thinly sliced shiitake mushrooms
1 cup thinly sliced onion
4 scallions, thinly sliced (about 1 cup)
½ cup peeled and thinly sliced carrot (about 1 medium carrot)
½ cup semisweet tonkatsu sauce (such as Bull Dog)
1 tablespoon ketchup
1 tablespoon soy sauce

Bring a medium saucepan 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.