To make apple cranberry chutney, combine butter, apples, cranberries, onion, brown sugar, ginger, orange zest, ½ teaspoon kosher salt, and ¼ teaspoon black pepper in a large saucepan.
Bring to a boil, reduce to a simmer, and cook until thickened, about 20 minutes. Pull off ½ of chutney and set aside. Strain remaining chutney through a fine-mesh sieve, pressing down with a rubber spatula to remove as much liquid as possible. Mix leftover syrup into reserved chutney, cover and save for serving.
Transfer strained chutney to a food processor and pulse until mostly smooth. Set aside.
Preheat oven to 350°F.
To butterfly the pork roast, lay the meat fat-side-down on a large cutting board. Use a sharp knife to cut lengthwise through the pork loin, ½-inch above the bottom of the meat, stopping ½-inch before the edge of the roast. Unfold the meat like the flap of a book and make another horizontal cut through the thicker half of the meat, ½-inch from the bottom and stopping ½-inch before the edge of the roast. Repeat making slices and unrolling the roast until it’s an even thickness of ½-inch when laid out. If necessary, cover butterflied pork with plastic wrap and gently pound with a meat mallet or rolling pin to an even ½-inch thickness. Discard plastic wrap.
Season meat all over with salt and pepper. Spread surface evenly with chutney and roll meat back into a tight spiral. Use butcher’s twine to tie the roll tightly in 1-inch intervals.
Heat butter and oil in a large cast iron pan over medium-high heat. When butter is melted, sear roast on all sides until evenly browned, 10-12 minutes total.
Place skillet in preheated oven and roast until an instant-read thermometer inserted into the center of the roast registers 135°F, about 45 minutes. Transfer to a cutting board, tent with foil, and rest 15 minutes. Remove twine, slice, and serve with leftover chutney.
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