Preheat the oven to 350 degrees F and line a baking sheet with parchment paper. Combine the sauerkraut and relish in a small bowl; set aside.
Unroll the crescent dough and pinch the seams together with your fingers. Cut the dough in half lengthwise, then slice each half crosswise into 12 strips. Spread about 1 teaspoon of the sauerkraut-relish mixture on each strip of dough, leaving a 1/2-inch border on one short end. Sprinkle with the cheese. Place 1 frank on each strip of dough and roll the dough around the frank, pressing lightly to seal.
Transfer to the baking sheet seam-side down; brush with the beaten egg and sprinkle with caraway seeds. Bake until the dough is golden brown, about 15 minutes. Serve with Thousand Island dressing and/or mustard.