Preheat the oven to 375 degrees F. Preheat an outdoor grill or indoor grill pan.
Score the duck skin diagonally along the breast, then crisscross by scoring the same way in the opposite direction. Season the duck with salt and pepper, to taste. In a large, oven-safe skillet over medium-high heat, add the olive oil. Put the duck breasts, skin side down first, into the pan and cook for 3 to 5 minutes on each side. Once they are browned on each side, transfer the skillet to the oven for 20 minutes.
In the meantime, bring the pineapple juice and rum to a boil in a medium saucepan, over medium heat. Reduce the mixture by half, then add the pink peppercorns, if using.
Grill the pineapple slices on both sides. Remove from heat to a cutting board. Cut pineapple into quarters if being used as an hors d'oeuvre. As a dinner course, cut however you like. Transfer the pineapple to a serving bowl and sprinkle with paprika.
Remove the duck from the oven. Pour some of the rendered duck fat and oil into the reduced pineapple rum sauce. Stir to incorporate. Slice the duck, arrange on a serving platter and serve with the pineapple. If serving as an hors d'oeuvre, cut the duck into small pieces.
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.