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Step 1
Score the duck breast skin. About an hour before cooking, remove the duck breasts from the refrigerator and their packaging. Use a sharp knife to score the skin: Make 3 to 4 parallel shallow cuts into the skin, but not through it, in one direction, then make 3 to 4 shallow cuts in another direction, making a cross hatch or diamond pattern. Pat the duck breasts dry with paper towels.
Step 2
Season with salt and rest for 15 minutes. Season the breasts on all sides with salt (you might not use all the salt; that’s fine). Set aside and let rest for 15 minutes. Meanwhile, cut the potato into small 1/2-inch cubes.
Step 3
Start in a cool pan to render out as much fat as possible. Place the duck breasts skin-side down in a large cast iron skillet and turn the heat to medium. You want to crisp the skin slowly, driving out as much of the fat as possible.
Step 4
Cook on the first side for 6 to 8 minutes. After 6 to 8 minutes, the pan will have filled with a substantial amount of fat from the skin. The skin should be a very dark golden-brown.
Step 5
Flip the breasts and cook 3 to 4 minutes, or until they reach 135°F. Flip the breasts and cook for 3 to 4 minutes more. Add the potatoes to the pan right after you flip the breast. Cook until the breasts are plump and register 135°F in the thickest part of the breasts for medium-rare.
Step 6
Rest the duck for 5 minutes and continue to cook the potatoes until crisp. Remove the duck breasts to a clean plate or cutting board and let rest 5 minutes. Continue cooking the potatoes, stirring regularly, until they are golden-brown on the outside and tender inside, 5 to 7 minutes.
Step 7
Slice the duck breasts and serve with the potatoes. Slice the duck breasts on the diagonal and serve over the potatoes with any accumulated juices on the resting plate.