Slice the fennel bulb in long, thin strips. Chop 3 tablespoons of the fronds and set fennel strips and fronds aside. In large saucepan, heat oil over medium heat. Add onion. Cook and stir till onion begins to brown. Add strips of fennel. Cook, covered, for 5 minutes or till fennel begins to wilt. Reduce heat to medium-low. In a large bowl combine apple and lemon juice. Add apple mixture to saucepan. Cook, covered, for 15 minutes, stirring occasionally. Stir in 1 tablespoon butter, caraway seeds, reserved fennel fronds, and cider or apple juice. Cook, uncovered, for 5 minutes more. Remove from heat; cool slightly. Season to taste with salt and pepper.(The sauerkraut can be made up to 2 days in advance and kept covered in the refrigerator till ready to use.)
Just before serving, warm sauerkraut, if necessary, and grill sausages. For a charcoal grill, grill sausages on the rack of an uncovered grill directly over medium coals for 3 to 7 minutes or until heated through, turning occasionally. Meanwhile, brush the insides of frankfurter buns or French rolls with melted butter. Toast on the grill rack until lightly browned. (For a gas grill, preheat grill. Reduce heat to medium. Place sausages, then buns or rolls on grill rack over heat. Cover and grill as above.) When ready to serve, place one sausage and a generous amount of the sauerkraut on the bun and serve with mustard on the side. Serve immediately. Makes 8 servings.