For the red-eye gravy: Combine the coffee, cane syrup, ham, vinegar and port in a saucepan. Cook over medium-high heat until reduced by half, then whisk in the butter. Strain through a fine-mesh sieve and set aside.
For the duck breast: Heat a medium skillet over high heat. Sprinkle the breasts all over with salt and pepper. Place the breasts, skin-side down, in the skillet and cook until the fat begins to render and the skin is golden brown, 1 to 2 minutes. Pour off the fat (reserve 2 tablespoons for the dressing). Reduce the heat to low and continue to cook, removing the fat as it renders, until the skin is tight and golden, 15 to 20 minutes. Increase the heat to medium-high, flip the breasts and cook until the meat is lightly browned but still medium-rare, 1 to 2 minutes more. Thinly slice the breast.
For the duck dressing: Heat a large cast-iron skillet over medium-high heat and add the 2 tablespoons reserved duck fat. Place the sprouts cut-side down in the skillet and cook until charred on the edges. Add the mushrooms and cook, stirring, 3 to 4 minutes. Fold in the pomegranate seeds and season with salt, pepper and lemon zest. Serve the duck with the dressing and gravy.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.