Bring stock and 2 cups water to a simmer over medium heat; add leeks and orzo and season with salt and pepper. Simmer until leeks are tender and orzo is al dente, about 8 minutes.
In a small heatproof bowl, whisk together egg yolks, lemon juice, and a pinch of salt. Slowly whisk in 1 cup stock, then gradually stir back into soup. Cook over medium heat, stirring often, until thickened slightly, 4 to 5 minutes. Stir in shredded turkey and watercress; adjust seasoning. Ladle soup into bowls and serve with more watercress, a crack of pepper, and a drizzle of oil.