Preheat the oven to 450 degrees F. Heat the olive oil in a large ovenproof skillet over medium-high heat. Season the chicken with salt and pepper and add skin-side down to the skillet. Cook until the skin browns, about 10 minutes. Flip the chicken, transfer the skillet to the oven and roast until cooked through, 10 to 12 minutes.
Meanwhile, trim the escarole, tear into pieces and rinse well. Thinly slice the garlic.
Remove the chicken to a plate. Discard all but 3 tablespoons fat from the skillet; return the skillet to medium-high heat. Add the garlic and cook until golden, about 1 minute. Add half of the escarole and cook, tossing, until wilted, about 1 minute. Add the remaining escarole and cook, tossing, until wilted, 1 minute; season with salt and pepper. Add the thyme and chicken broth and bring to a simmer. Cook until reduced by half, about 3 minutes. Add the capers, butter and lemon juice. Cook, swirling the pan, until the butter is melted. Discard the thyme sprigs.
Divide the chicken thighs and escarole mixture among plates. Sprinkle with parsley.