In a large saucepan or pot, bring the leeks, potatoes, fennel, thyme, broth, vinegar, salt, and pepper to a boil. Reduce heat to medium-low and simmer gently for 5 minutes. Add the chicken, return to a gentle simmer, and cook for 7 minutes. Add the Swiss chard or kale and cook until the greens are tender and the chicken is cooked through, about 3 minutes. Transfer the chicken to a cutting board and thickly slice it. Divide the chicken among individual bowls. Spoon the potato and fennel mixture over the chicken and drizzle with the oil.
Tip: Poaching means simmering something gently, allowing only an occasional bubble to come to the surface. For this dish, the difference between poaching and boiling means supple rather than rubbery chicken and tender vegetables instead of mushy ones.