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Mix ½ cup sugar and 3 tablespoons salt; rub mixture precise via pork. Location pork in a gargantuan bowl, quilt with plastic wrap, and sit back for 8 to 12 hours.
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Preheat oven to 300°F. Switch pork, plump aspect up, to a heavy roasting pan, discarding any liquid in bowl. Roast unless very relaxed and a thermometer inserted into thickest half registers 145°F, 4 to 4½ hours. Switch pork to a platter and let relaxation unless cool adequate to take care of, about 30 minutes. Pull into gargantuan objects, discarding the bone.
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Mix radishes and carrots in a heatproof bowl. Bring vinegar, 1 cup water, and final ¼ cup sugar and 1 tablespoon salt to a boil in a medium saucepan over high. Cook dinner, stirring consistently, unless sugar and salt dissolve, about 4 minutes. Pour hot brine over radishes and carrots and build of abode apart unless cool, about 30 minutes. (Pickled greens can even be made 2 days forward; refrigerate in an airtight container.)
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Salvage lettuce wraps with pork, greens, and kimchi.