Score the duck skin in a cross-hatch pattern. Season with salt and pepper. Heat a large cast iron skillet. Add the duck breasts, skin side down, and cook over moderately low heat until most of the fat has been rendered, for about 8 minutes. Increase the heat to moderate and cook the duck breasts until the skin is browned and crisp, about 4 minutes. Turn the duck breasts and cook them until they are medium-rare, about 3-4 minutes longer. Transfer the duck breasts to a carving board, skin side up, and let them rest for 5 minutes.
Reserve four tablespoons of duck fat from the skillet. In a bowl, stir the duck fat with the lime zest, lime juice, Tabasco, honey and garlic and season lightly with salt. Add cucumbers.
Slice the duck breasts crosswise and transfer to plates. Spoon the relish around the duck and swirl with jam. Serve with steamed rice.
Aleksandra aka QueenSashy is a scientist by day, and cook, photographer and doodler by night. When she is not writing code and formulas, she blogs about food, life and everything in between on her blog, Three Little Halves. Three Little Halves was nominated for 2015 James Beard Awards and the finalist for 2014 Saveur Best Food Blog Awards. Aleksandra lives in New York City with her other two halves, Miss Pain and Dr. V.