Season chicken with salt and pepper and dust with a light coating of flour, shaking off any excess.
Heat 1 tablespoon of the butter along with the olive oil in a large non-stick skillet set over medium-high heat. Add chicken and cook for 3–4 minutes per side, or until golden and cooked through. Remove chicken from pan and set aside.
Add remaining 1 tablespoon butter, lemon juice, chicken broth, capers, garlic, Italian seasoning, and paprika (if using) to the skillet. Return skillet to medium heat and bring to a boil while scraping down the brown bits.
Add kale along with the cooked chicken back to the skillet. Cook for an additional 2 minutes or until the kale has wilted and the chicken is heated through.