To prepare Brussels sprouts, remove any yellow or brown outer leaves, cut off the stems, and cut in halves or quarters.
Check every 10 minutes, turning frequently with a spatula to ensure even roasting, until Brussels sprouts are fork tender.
Meanwhile, in a small saucepan, cook and stir apple juice and maple syrup over medium heat until it comes to a simmer.
Let simmer, uncovered, for 10 to 15 minutes, or until it reduces to a syrup-like consistency.