Combine the apple cider, beef stock, cinnamon stick, cloves, peppercorns, and shallot in a 2-quart saucepan, and bring to a boil over medium-high heat. Reduce the heat so that the liquid just simmers, and cook until the mixture, including the seasoning, has reduced to 1 cup, 35 to 40 minutes. (Have a liquid measure nearby so that you can check the volume as necessary.) Let the sauce cool for a few minutes; then strain it through a fine-mesh sieve into a bowl, discarding the shallot and spices. Set it aside.

Season the duck breasts on both sides with the salt and pepper. Heat the olive oil in a 12-inch skillet over medium heat. Add the breasts, skin side down, and cook for 2 minutes. Then reduce the heat to medium-low and cook for 7 minutes longer, or until the skin is golden brown and crisp and most of the fat has been rendered. Carefully drain off most of the fat. Turn the duck breasts, skin side up, raise the heat to medium, and cook for 3 to 4 minutes for medium-rare. Transfer the duck breasts to a cutting board, and allow them to rest for 5 minutes.

Drain and discard any remaining fat from the skillet, and set it over medium-high heat. Add the sauce and cook until it has reduced by half or until it is syrupy, about 4 minutes. Remove from the heat.

Slice the duck breasts diagonally into 1/4-inch-thick slices. Serve with the sauce spooned over the top.