Place the carrots in a small pot. Add the apple juice, bay leaves, brown sugar, and cinnamon. Stir to combine. Cover and bring to a boil over high heat, then lower the heat to medium and remove the cover. Cook at a low boil, uncovered, for about 4 minutes, until just fork-tender. Stir in the vanilla, and add salt and pepper to taste. Taste for seasonings.
Using a slotted spoon, transfer the carrots to a shallow bowl. Raise the heat to medium-high and bring the cooking liquid to a boil. Cook at a medium boil for about 15 minutes, until the liquid reduces into a syrupy consistency. Remove the bay leaves. Pour this liquid over the carrots.