Melt butter in a large skillet over medium heat; cook and stir apples until lightly browned, 5 to 10 minutes. Sprinkle brown sugar over apples and cook until apples are tender, about 5 minutes more. Transfer apples to a plate.

Arrange chicken on a sheet of waxed paper; season both sides with cinnamon, salt, and pepper. Melt remaining butter in the same skillet and cook chicken until browned, 2 to 3 minutes per side. Transfer chicken to a plate.

Cook and stir onion in the same skillet used for chicken until tender, 5 to 10 minutes. Stir cider and vinegar into onion, reduce heat to medium-low, and simmer until sauce is slightly thickened, about 2 minutes.

Return chicken to onion mixture, spooning cider sauce over chicken and cook until chicken is no longer pink in the center and the liquid is reduced by half, 5 to 10 minutes more. Spoon apples into chicken mixture and cook until apples are warmed, 2 to 3 minutes.

Arrange noodles on plate, top with chicken, and spoon sauce over chicken.