1. Combine chicken, broth, 1 cup leek greens, and 2 cups water in a pot. Season with salt and pepper. Bring to a boil, then reduce heat and simmer until chicken is cooked through, 15 minutes. Transfer chicken to a plate; strain liquid through a fine-mesh sieve. (You should have about 5 cups.) Rinse pot; return strained broth to it and place over medium heat. Add barley and leek rounds; bring to a simmer and cook until leeks are tender, 6 to 8 minutes. Meanwhile, shred chicken into bite-size pieces (discarding skin and bones).

2. Whisk together yolks, lemon juice, and a pinch of salt in a heatproof bowl. Slowly add 1 cup hot broth to yolk mixture, whisking constantly. Remove broth from heat; slowly stir yolk mixture back into broth. Return chicken to broth; cook over medium heat, stirring often, 5 minutes (do not let boil). Stir in watercress; season with salt and pepper. Spoon soup into a bowl; top with more watercress, pepper, and a drizzle of oil. Serve.