Preheat oven to 400°.
Slice the ends off of the Brussels sprouts, drizzle with a little olive oil, and season them with salt, pepper.
Cut the Dumpling Squash in half, or the Delicata Squash into 1"-thick rings. If using Dumpling Squash, fill the center cavities with maple syrup and butter, If using Delacata, brush lightly with melted butter or olive oil.
If you don't have pre-cooked chicken, you can also place a raw chicken breast on the same sheet pan, drizzled with oil oil, and salt and pepper, and roast with the sprouts and squash.
Bake in oven for approximately 30 minutes, checking to make sure chicken is no longer pink in the middle and squash and sprouts are tender.
Whisk the salad dressing ingredients together in a bowl, or shake in a pint-sized mason jar.
Assemble the salad by creating a bed of arugula, adding the chopped figs, dried cranberries, and seeds and pecans.
When chickena and veggies are done, and have cooled enough to handled, slice the chicken and add to top of salad. Slice the Dumpling Squash into rings, and add to salad. Add Brussels sprouts to salad whole.
Sprinkle with shredded cheese and drizzle dressing over the top of all.