Chef John's Hot and Sour Soup

Name: Chef John's Hot and Sour Soup
Prep time: 20 min
Cook time: 30 min
Total time: 50 min
Cuisine: Chinese
Ingredients: 0.75 ounce dried shiitake mushrooms; 2 tablespoons soy sauce, or more to taste; 0.25 cup seasoned rice vinegar; 1 teaspoon ground white pepper, or to taste; 0.25 teaspoon sesame oil; 3 tablespoons cornstarch, or as needed; 3 tablespoons water, or as needed; 2 teaspoons vegetable oil; 1 teaspoon minced fresh ginger root; 0.25 cup sliced green onions; 4 cups chicken broth; 0.25 cup grated carrot; 2 tablespoons thinly sliced red bell pepper; 0.5 cup bamboo shoots; 1 cup cubed tofu; 2 large eggs, beaten