Sesame oil, for drizzling
3 scallions, chopped
5 cups chicken broth
2 packages fresh thin wonton noodles
1/4 tsp. white pepper
Kosher salt
1 tsp. Shaoxing wine
12 oz. chicken breast
1 1/2 tsp. cornstarch, divided
2 tsp. vegetable oil, divided
1 tsp. sugar
2 1/2 cups water
16 dried shiitake mushrooms