Using a sharp knife, trim away excess skin from the duck (leaving enough to amply cover the breast). Using a sharp knife, score the skin, first cutting diagonally in one direction, and then the other, in a crosshatch manner. Cut all the way through the skin and most of the fat, but avoid the flesh. Alternatively, you can use a Jaccard tool to pierce the skin. On a small rimmed baking sheet or a plate, place a 1/4 inch layer of salt rough the size of the duck breast. Place duck breast skin-side down on bed of salt. Let stand at room temperature for 1 hour. Remove and brush off salt with a stiff brush. Line a small rimmed baking sheet or shallow dish with ice cubes and cover with plastic wrap. Place duck breast skin-side down on plastic wrap and weight it with a cheesecloth bundle filled with pie weights or dried beans for 25 minutes at room temperature.
Preheat oven to 350 degrees. Remove cheesecloth bundle and season flesh side with pepper, and place skin side down, in an unheated 10-inch skillet. Top again with cheesecloth bundle and cook over medium-low heat until a small pool of fat forms in the pan. As fat accumulates, spoon off into a heatproof bowl and reserve for another use; allowing it to cool before storing in an airtight container at room temperature. Continue to cook duck until the skin is nicely browned and crisp, about 25 minutes. Use tongs to turn breast over and top with cheesecloth bundle for 1 minute. Remove bundle and transfer skillet to oven and cook until duck is medium rare, 8 to 12 minutes. It should register 125 degrees on an instant-read thermometer inserted into the thickest part. Transfer to a wire rack set over a rimmed baking sheet to rest for 5 to 7 minutes. The duck will continue to cook slightly during this time.
Drain all of fat from the pan and place over medium heat. Add the shallots and cook until soft, about 2 minutes. Increase heat to medium high and add the port, chicken stock, and thyme, scraping the bits from the bottom of the pan with a wooden spoon. Bring to a simmer and cook until sauce is syrupy and reduced to 1/3 cup. Remove from heat and whisk in butter until fully incorporated. Season to taste with salt and pepper and serve.
Slice duck crosswise on a slight angle into 1/4-inch-thick slices, fan out on serving plates, and drizzle with sauce before serving.