How to Make It
Combine ½ cup sugar and 3 tablespoons salt; rub mixture all over pork. Place pork in a large bowl, cover with plastic wrap, and chill for 8 to 12 hours.
Preheat oven to 300°F. Transfer pork, fat side up, to a heavy roasting pan, discarding any liquid in bowl. Roast until very tender and a thermometer inserted into thickest portion registers 145°F, 4 to 4½ hours. Transfer pork to a platter and let rest until cool enough to handle, about 30 minutes. Pull into large pieces, discarding the bone.
Combine radishes and carrots in a heatproof bowl. Bring vinegar, 1 cup water, and remaining ¼ cup sugar and 1 tablespoon salt to a boil in a medium saucepan over high. Cook, stirring constantly, until sugar and salt dissolve, about 4 minutes. Pour hot brine over radishes and carrots and set aside until cool, about 30 minutes. (Pickled vegetables can be made 2 days ahead; refrigerate in an airtight container.)
Build lettuce wraps with pork, vegetables, and kimchi.