Peel off any wilted or discolored leaves from the Brussels sprouts. Beginning with the top, slice the Brussels sprouts as thinly as possible (a mandoline made quick, easy work of this), discarding the stem end.
Put the Brussels sprouts in a large bowl. Add the red cabbage and carrot and fluff with your fingers until the vegetables are thoroughly mixed.
In a small bowl, mix the lime juice, flax oil, vinegar, cashew butter, miso, tamari, garlic, sriracha, and ginger in a small bowl and whisk until smooth and well combined. Pour over the vegetables and toss well.