1. Place duck breast halves between two sheets of plastic wrap. Pound lightly to an even thickness, about ½–¾ inch. Discard plastic wrap. Using sharp knife, score skin in a ¾-inch crosshatch pattern, being careful not to cut into the meat. Cover and chill. This can be done up to 8 hours before cooking.
2. In a large, heavy skillet, melt 1 tablespoon butter over medium-high heat. Season duck with salt and pepper. Add duck, skin side down, to skillet and cook until skin is browned and crisp, about 5 minutes. Turn duck over, reduce heat to medium, and cook until browned and cooked to desired doneness, about 4 minutes longer for small breasts and 8 minutes longer for a large breast for medium rare. Transfer to work surface, tent with foil to keep warm, and let rest 10 minutes.
3. Meanwhile, pour off all but 2 tablespoons of drippings from skillet. Add shallot and stir over medium heat for 30 seconds. Add broth, cherries, tawny port, and honey. Increase heat to high; stirring often, boil until sauce reduces to a glaze, about 3 minutes. Whisk in 1 tablespoon cold butter. Season sauce to taste with salt and pepper.