Hack off the knobs from the ends of the chicken legs with a heavy knife. Blot the excess moisture from the chicken with paper towels. Season the chicken with salt, black pepper and cayenne pepper. Dust the pieces lightly with flour. Melt the duck fat in the fryer. Place the chicken pieces into the fat. Cover the fryer. Fry until the skin is brown - about 15 minutes.
Slice the potatoes into 1/2 inch thick half-moon shapes.
Turn the chicken pieces over. Add the potatoes, making sure that all of the potatoes have contact with the bottom surface of the pan. Salt and pepper the potatoes to taste. Cover the fryer.
Continue turning the potatoes until they are golden-brown and finished. Remove the chicken and potatoes to a platter. Arrange the potatoes around the chicken. De-grease the fryer. Pour in the wine, scrape the bottom of the pan and reduce the liquid (but not to a glaze). Whisk in the cold butter - a piece at a time. When the butter is incorporated, pour the sauce over the chicken, immediately.