Soak dried shiitake mushrooms in hot water, stirring occasionally, until soft, 15 to 20 minutes. Drain well and slice into thin strips.
Whisk soy sauce, vinegar, ground white pepper, and sesame oil together in a small bowl for hot and sour mixture; set aside until needed.
Whisk cornstarch and water together in another small bowl for slurry; set aside until needed.
Heat vegetable oil in a saucepan over medium-high heat. Add ginger, sliced green onions, and shiitake mushrooms. Cook, stirring, for about 2 minutes. Pour in chicken broth and bring up to a simmer. Reduce heat to medium and simmer for 5 minutes.
Stir in carrot, red pepper, bamboo shoots, tofu, and the hot and sour mixture. Let simmer for 5 minutes.
Raise heat to medium-high and bring to a boil. Stir the soup slowly with a spoon while transferring in the beaten eggs in a steady stream to create thin ribbons. Keep stirring until the soup comes back to a boil. Reduce heat to medium.
Stir the cornstarch slurry to make sure it's well blended, then slowly drizzle it into the soup while stirring constantly. Soup should thicken up within 1 to 2 minutes. Turn off heat, taste for seasoning, and serve.