Combine the bread crumbs, paprika, salt, and pepper in a plastic bag. Place chicken breasts in the bag, and shake to coat.
Melt butter in a large skillet over medium heat. Add chicken breasts, and cook for 4 minutes on each side, or until brown.
Remove chicken to a plate, and keep warm. Pour the cola into the skillet, and stir to remove any browned chicken bits from the pan. Add ham, mushrooms, sherry, and balsamic vinegar. Reduce heat to medium-low, and simmer, uncovered, until the liquid has reduced by 1/2.
Stir sour cream into the skillet, and return chicken breasts to the pan. Cover, and simmer for 5 more minutes, or until chicken is no longer pink, or the internal temperature is 170 degrees F (75 degrees C). Remove chicken to serving plates, and spoon sauce over.