In a large saucepan or pot, bring the leeks, potatoes, fennel, thyme, broth, vinegar, ½ teaspoon salt, and ¼ teaspoon 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 vegetables over the chicken and drizzle with the oil.