Heat the oil in a large skillet over medium heat until shimmering. Liberally season the chicken breasts on both sides with salt and pepper. Sear the chicken on one side until browned, about 5 minutes.
Place the potatoes in a 6-quart or larger slow cooker. Sprinkle with the 1/2 teaspoon salt. Place the chicken breasts in the slow cooker, seared-side up. Pour in the broth, lemon juice, and capers.
Cover and cook on the LOW setting until the chicken is cooked through, tender, and registers an internal temperature of 165°F, 3 1/2 to 4 hours. Meanwhile, use your fingers to rub the flour into the butter; set aside.
Transfer the chicken and potatoes to a serving dish. Pour the liquid and capers into a medium saucepan. Bring to a boil over medium heat. Add the butter mixture, whisking constantly until it's completely incorporated into the sauce. Cook, stirring frequently, until the sauce is thickened, 3 to 4 minutes. Serve over the chicken and potatoes, garnished with the parsley.