Combine the broth and garlic in a large pot over medium heat. Season with a generous pinch each of salt and pepper and bring to a boil. Stir in the kale in large handfuls, letting each batch wilt slightly before adding more. Cover and cook over medium heat, stirring occasionally, until the greens are just tender, about 10 minutes.
Meanwhile, place a chicken cutlet between two sheets of plastic wrap and lightly pound until very thin; transfer to a plate or baking sheet. Repeat with the remaining chicken cutlets. Season one side of the cutlets with salt and pepper. Place 2 sage leaves, 1 slice prosciutto and 1 tablespoon chopped peppers on one half of each cutlet. Fold each cutlet over the filling. Gently press down to help the stuffed cutlets adhere and stay folded during cooking. Season the outside of the cutlets with salt and pepper.
Heat the olive oil in a large skillet. Add the stuffed cutlets and cook over moderately high heat, turning once, until golden and just cooked through, 6 to 8 minutes total.
Transfer the greens to a platter with tongs or a slotted spoon. Put the chicken on top of the greens and tent the platter with foil. Ladle the broth from the large pot into the large skillet and bring the liquid to a boil. Add the remaining 1/4 cup pickled peppers and cook, scraping up any browned bits, until the sauce is slightly reduced, 2 to 3 minutes. Spoon the sauce over the chicken and greens before serving.