In a small bowl, combine the paprika, salt, and pepper. Pat the chicken dry and season all over with the paprika mixture.
Heat the olive oil in a large skillet or braiser over medium-high heat. Once the oil is glistening, working in batches, add the chicken skin sides down. Cook until the skin is golden brown and crisp, about 5 minutes. Flip and continue cooking until just browned, about 3 minutes more. Transfer to a plate and continue with the remaining chicken.
Once the chicken is done, reduce the heat to medium. Add the shallot and cook, stirring often, until just tender, about 4 minutes. Add 1 cup of the chicken stock and the mustard. Bring to a simmer, scraping up any brown bits as you stir, until the sauce is reduced and thick enough to coat the back of a wooden spoon, about 5 minutes.
Remove the skillet from the heat and slowly whisk in the cold butter. Once the butter is melted, stir in the lemon juice, tarragon, and capers. Add the kale and beans and toss until coated and the kale is wilted. Return the chicken to the pan, nestling it in.
Pour the broth over the chicken, cover, and bring to a simmer over low heat. Cook until the internal temperature of the chicken reaches 165°F on a meat thermometer, about 35 minutes.
