Score the duck breasts through the skin and fat every 1/2-inch, taking care not to cut into the meat. Sprinkle the duck on both sides with salt. Place skin-side down in a medium skillet and then raise the heat to medium. As the pan heats, the fat will render. Cook until the skin is a deep golden brown, about 12 minutes, spooning off some of the fat as it cooks, and reserving a tablespoon or so. Flip the duck and cook until an instant read thermometer registers between 128 and 132 degrees F for medium rare, another 4 to 6 minutes. Transfer to a cutting board to rest. While the duck is cooking, trim the thickest stems from the purslane and discard. Tear the purslane into 3- to 4-inch pieces. Chill until ready to serve. Halve the grapes and set aside.
When the duck is done, remove all of the fat from the pan and reduce the heat to medium-low. (If there are little brown bits stuck to the pan, just wipe or scrape them out.) Return 2 teaspoons of duck fat to the pan. Add the grapes and stir for about 2 minutes just to warm them. Add the vinegar, and stir for about a minute, scraping the bottom of the pan. Season with salt and pepper.
Pour the grape mixture over the purslane and toss. Season with salt and a generous amount of black pepper. Divide the salad among four plates. Thinly slice the duck breast and place 4 or 5 slices, overlapping slightly, alongside each salad. Serve immediately.
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