Remove the bottoms of the brussels sprouts, then using a mandoline, processor, or good knife, thinly slice them.
For a trick on how to remove the seeds from a whole pomegranate, halve the fruit and while submerged in a bowl of water, separate the seeds from the membrane. Scoop the pieces of membrane from the water and drain the seeds.
Toss the sliced brussels sprouts in a large bowl with the pomegranate seeds, chopped mint, and walnuts.
In a separate bowl, whisk together all the ingredients for the dressing and toss with the salad. Season with salt and pepper.