Clean and cut the fennel in long strips (julienne) and trim tops, reserving for later use.

Heat oil in heavy-bottomed saucepan over medium heat, add onion and kosher salt. Cook until onion begins to brown, add strips of fennel, stir, and let cook covered for about 5 minutes or until fennel begins to wilt, reduce heat to medium-low.

Meanwhile, mix grated apple with lemon juice and add to pan, cover and cook, stirring occasionally for 15 minutes. Add butter, mixing well. Add caraway seeds, reserved fennel tops and hard cider or apple juice. Cook for an additional 5 minutes, uncovered. Season, to taste, with salt and pepper. Remove from heat and let sit to allow the flavors to mingle. (The sauerkraut can be made up to 2 days in advance and kept covered in the refrigerator until ready to use.)

Build charcoal fire or preheat gas grill. Just before serving, warm sauerkraut and grill sausages. To grill sausages, place on the cooking grate over direct medium heat for 6 to 8 minutes, turning occasionally to mark all sides of the sausage. Meanwhile, split the buns and brush a little butter on the inside. Toast until lightly browned. When ready to serve, place 1 sausage and a generous amount of the sauerkraut on the bun and serve with mustard on the side. Enjoy immediately.