If the chicken breasts are thick, slice them in half horizontally to make thinner, even pieces.
In a small bowl, mix the brown sugar, 1 tablespoon olive oil, garlic, salt, pepper, and paprika. Rub the mixture over both sides of the chicken.
Heat the remaining 1 tablespoon olive oil in a large skillet over medium heat. Add the chicken and cook for 4 to 5 minutes per side, or until the thickest part reaches an internal temperature of 165°F. Transfer the chicken to a plate and loosely cover with foil.
In the same skillet, melt the butter over medium heat. Add the apples, cranberries, apple cider, brown sugar, apple cider vinegar, cinnamon, and nutmeg. Stir to combine.
Cook for 10 to 12 minutes, stirring occasionally, until the apples are tender, the cranberries have burst, and the sauce has thickened.
Return the chicken to the skillet and spoon the cranberry apple topping over each piece. Let everything warm together for 1 to 2 minutes, then remove from the heat and let the chicken rest for 5 minutes before serving.