In a large saucepan, over high heat, bring the stock, water, soy sauce, mushrooms, ginger and red pepper flakes to a boil. Lower the heat simmer, covered for 10 minutes or until the mushrooms are tender. Blend the sesame, vegetable oils and rice vinegar. Toss this dressing with the bean sprouts, scallions and watercress and season to taste with salt and pepper. Divide this salad among 4 soup bowls and reserve for later. When the mushrooms are tender, add the chicken to the soup and simmer, over moderate heat, uncovered for 5 minutes or until the chicken is cooked through. Season to taste with salt and pepper. Remove the ginger slices, then ladle the hot soup over the bean sprout mixture and garnish with sesame seeds.