Yakisoba (Japanese Stir-Fried Noodles)

Ingredients

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
1/4 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)
1/2 cup peeled and thinly sliced carrot (about 1 medium carrot)
1/2 cup semisweet tonkatsu sauce (such as Bull Dog)
1 tablespoon ketchup
1 tablespoon soy sauce