Hot-and-Sour Seafood Soup

1 tablespoon vegetable oil

1 bunch scallions; whites cut into 1/4-inch pieces,greens cut into 2-inch pieces

1 tablespoon minced peeled ginger

1 tablespoon low-sodium soy sauce

1/2 teaspoon sugar

Kosher salt and freshly ground pepper

4 ounces shiitake mushrooms, stemmed and sliced

1/4 cup cornstarch

2 plum tomatoes, cored and cut into large chunks

12 ounces firm white fish (such as pollock), cut into 1-inch pieces

6 ounces bay scallops

2 tablespoons balsamic vinegar