Pre-heat oven to 450°F.
Trim stem ends from Brussels sprouts and remove any browned or wilted outer leaves.
Halve Brussels sprouts lengthwise through the core.
Thinly slice ¼ of the halved Brussels sprouts; set aside as these are not roasted.
On a rimmed baking sheet, toss halved Brussels sprouts with 3 tablespoons oil.
Season with ¾ teaspoon salt and spread into a single layer.
Place in oven and roast, stirring once halfway through, until browned and crisp in spots, about 20 minutes
Meanwhile, in a medium skillet, combine apple juice, apple cider vinegar and 2 tablespoons Dijon mustard; season with ½ teaspoon salt and a few grinds pepper.
Bring to a boil over high, reduce to medium and cook, stirring occasionally until thickened and reduced to about ½ cup, 12 to 15 minutes.
In a large bowl, place raw sliced Brussels sprouts.
Add cider syrup and 2 tablespoons oil and toss to coat.
Score pomegranate around the perimeter – do not cut all the way through – and pull apart.
Hold one pomegranate half in one hand, cut side down, just inside a large pot and, using a large spoon, whack the outside repeatedly until seeds fall out.
Repeat with the other pomegranate half.
Add roasted Brussels sprouts and half of the pomegranate seeds to bowl with the sliced Brussels sprouts.
Season with ½ teaspoon salt and a few grinds pepper and toss to combine.
Transfer to a serving bowl, or platter, and top with more pomegranate seeds.
