In a medium saucepan over high heat, bring the apple juice to a boil, stirring occasionally. Continue boiling until the juice is syrupy and reduced to about ⅓ cup, 20 to 30 minutes. Remove from the heat and stir in the mustard. Set aside. If the glaze hardens before you need to use it, gently reheat it (or keep it warm on a very low burner).
Meanwhile, prepare the sauerkraut. In a large sauté pan over medium-high heat, warm the oil. Add the garlic and cook until fragrant, about 1 minute. Add the remaining sauerkraut ingredients, except the apple, cover, and cook for 5 minutes. Uncover, stir in the apple, replace lid, and cook for 5 minutes. Uncover, and cook until the sauerkraut is tender and the liquid has been absorbed, 5 minutes, stirring frequently. Remove from the heat.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Lightly brush the sausages with oil, and then grill over direct medium heat, with the lid closed, until heated through, about 6 minutes, turning and basing with the glaze two times (don’t use all of the glaze—you will need some to baste the apple slices). Remove from the grill and let rest for 3 to 5 minutes.
Lightly brush the apple slices on both sides with oil, and then grill over direct medium heat, with the lid closed, until slightly softened, 2 to 4 minutes, turning and basting with the glaze once. Serve the sausages warm with sauerkraut and apple slices.