Chinese Chicken Noodle and Egg Drop Soup

Heat the oil in a medium saucepan over medium high heat until hot. Season the chicken with 1/4 teaspoon each salt and pepper, then sear the chicken, turning occasionally, until browned and cooked through, about 8 minutes. Transfer the chicken to a plate and let cool to warm, then shred the chicken. Bring the stock, ginger, and 1 teaspoon salt and 1/2 teaspoon pepper to a boil in a the same saucepan. Stir in the noodles and cook until tender, about 5 minutes. Stir together the soy and corn starch, then whisk into the broth and boil until the soup is slightly thickened, 1 to 2 minutes. Remove the saucepan from heat. Whisk together the eggs, sesame oil, and 1/2 teaspoon salt, then pour the eggs into the hot broth in a slow stream, whisking constantly to scatter the eggs as they cook. Stir in the chicken, then season the soup with salt to taste. Serve the soup topped with the scallions.