In a small saucepan set over medium heat, add the olive oil and when it is warm add the garlic and ginger and cook until the garlic is fragrant but not brown, about 30 seconds. 

Add the chicken stock, rum and ginger preserves, bring to a boil stirring to melt the preserves. Lower the heat to medium low and cook, reducing the sauce to a thick sauce, about 20-25 minutes. Stir in the cream and add salt and pepper to taste. Cover and reserve while preparing the duck breast.

Preheat the oven to 425°F

Score the skin on each duck breast, diagonally, turn and slice again in a crisscross pattern. Sprinkle each breast on the skin side with salt and pepper, rubbing gently, working the salt into the cuts. Sprinkle the underside with salt and pepper.

Heat a cast iron skillet or other heavy oven-proof skillet over medium-high heat until hot but not smoking. Add the duck breasts, skin side down. Cook until the skin is well browned and the fat has rendered, about 5 minutes. You will see a lot of fat bubbling out around the breasts. Pour off the fat, flip the breasts to skin side up and transfer to the middle of the oven. Roast the duck breast until they have reached an internal temperature of 135°F on an instant read thermometer, about 8 minutes. *(See Notes)

Remove the skillet from the oven and transfer the duck breasts to a cutting board, tent with foil and let rest for 5-minutes, then slice diagonally.

To serve, spoon a puddle of the sauce onto each of two warm plates and arrange the slices of one duck breast on each plate.