Pat chicken dry with paper towels, then season all over with salt and pepper. Heat oil in a large Dutch oven over medium-high heat. Add half the chicken skin-side down and sear 5 to 7 minutes, until deep golden brown. Transfer to a plate and repeat with the remaining chicken; set aside.

Add garlic, onion and thyme to the same pan and cook about 2 minutes, stirring frequently, until garlic is light golden brown. Add 1/2 cup broth; scrape up any browned bits from the bottom of the pan and cook about 2 minutes, until almost completely evaporated. Add remaining 1 1/2 cups broth, then return chicken skin-side up and any chicken juices to the pan. Vigorously simmer 40 to 45 minutes, uncovered, over medium heat until only about 1/4-inch liquid remains and chicken is very tender, no longer pink inside and reaches an internal temperature of 165°F.

Remove chicken to a clean plate and cover loosely with foil to keep warm.  Add kale to the pan a few handfuls at a time, stirring until wilted. Stir in vinegar or lemon juice. To serve, place chicken on bed of greens; spoon sauce over chicken.