Bring the chicken broth, ginger and scallions to a boil in a saucepan. Cover, reduce the heat to medium low and simmer 20 minutes. Remove from the heat and strain into a large bowl; discard the ginger and scallions. Return the broth to the saucepan and stir in the mirin. Meanwhile, toss the chicken, sesame oil, 1 tablespoon cornstarch, 1/2 teaspoon salt and 1/2 teaspoon white pepper in a bowl. Refrigerate until ready to use. Dissolve the remaining 3 tablespoons cornstarch in 1/4 cup water. Return the broth to a simmer over medium heat. Add the cornstarch mixture and cook, stirring, until glossy and slightly thickened, about 4 minutes. Add the chicken and simmer until cooked through, 1 to 2 minutes. Drizzle the eggs into the soup a little at a time, creating thin sheets of cooked egg. Gently stir in the watercress. Divide the soup among bowls.