Heat 2 teaspoons of the oil in a large skillet over medium heat. Add the kale, garlic and 1/4 teaspoon salt and cook, stirring, until the kale is softened and the garlic is fragrant, about 5 minutes. Add the corn and cook, stirring, until warmed through, 2 to 3 minutes more. Transfer to a bowl to cool slightly, and then stir in the cheese.
Insert a thin paring knife into the thickest part of the chicken breast and cut down the side to make a 3-inch pocket. Repeat with the remaining chicken breasts. Then evenly stuff with the kale mixture.
Heat a large nonstick skillet over medium-high heat until very hot, 3 to 4 minutes. Rub the chicken breasts with the remaining 2 teaspoons oil and sprinkle with a total of 3/4 teaspoon salt and 1/4 teaspoon pepper. Add all 4 chicken breasts to the skillet and cook until golden brown, about 2 minutes. Then lower heat to medium. Continue cooking, turning once halfway through, until just cooked through, about 14 minutes more. Insert a small paring knife into the top of the chicken to test for doneness. If any pink areas remain, cover the skillet and continue to cook until opaque.
Transfer the chicken and any bits of filling that fell into the skillet onto 4 plates. Stir the broth and flour together in a small bowl and then add to the skillet. Cook, stirring, until thickened, about 2 minutes. Remove from the heat, stir in the lemon juice and season with salt and pepper. Spoon the pan sauce over the chicken.
Recipe courtesy Food Network Kitchens