Instructions

• In a medium saucepan over high heat on the stove, bring the apple juice to a boil, stirring occasionally. Continue boiling until the juice is syrupy and reduced to about 80 milliliters, 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 skillet over medium-high heat on the stove, 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 is absorbed, about 5 minutes, stirring frequently. Remove from the heat. 

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• Prepare the grill for direct cooking over medium heat ( 180° to 230°C).
• 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 sausages with oil, and then grill over 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.
  Lightly brush the apple slices on both sides with oil, and then grill over 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.