Preheat the oven to 400F and line a baking sheet with parchment. Place the squash on the baking sheet and drizzle it with oil. Sprinkle it with salt and pepper, then use your hands to mix everything. Transfer the squash to the oven and roast for 35 minutes, or until lightly browning at the edges. Alternately, you can use an air fryer and oil spray to cook the squash: 12-13 minutes at 375F, shaking the basket once through.
To make the dressing, blend all ingredients together in a high speed blender till smooth. This should be a relatively thick dressing, but if it's too thick to pour into a jar or a storage container, add a few extra tablespoons of water to thin it. You'll have about 1 cup dressing and will probably use 1/2-2/3 cup for the salad. The leftover dressing can be stored in an airtight container for up to 5 days.
When the squash is ready, toss it together with the shaved Brussels sprouts and pomegranate arils. Add about 1/2-2/3 cup of the dressing to the salad. Toss to combine, then season to taste with salt and pepper. Serve.
Leftover salad will keep for a day in an airtight container in the fridge.