Wash and scrub carrots. Slice into rounds. Transfer to a large pot.
Add the apple juice. Bring liquid to a simmer, reduce heat to medium-low, and simmer an hour or until carrots are fork tender.
While carrots are cooking, the juice will reduce (this is desired); however, watch to make sure enough liquid remains to keep the carrots covered. If liquid level gets too low, simply add water.
Once carrots are tender, add the brown sugar to the pot. Once the sugar is dissolved, remove from the heat.
Transfer the carrots to hotel pans with enough juice to keep the carrots moistened. Cover with foil and keep warm in a 200℉ oven. Alternatively, refrigerate if preparing in advance to reheat later.