Lightly season chicken with salt and pepper. Working in batches, sauté chicken in olive oil and a large Dutch oven until lightly golden. Remove from heat and set aside.
In the bowl of a large food processor, add walnuts. Process until the walnuts form a thick chunky paste (you may need to scrape the sides a couple times). Add a few tablespoons of cold water through the feed chute while walnuts are still being processed. Continue processing until paste becomes beige in color throughout.
In a large Dutch oven, heat oil over medium high heat. Add onions and garlic and cook for 1-2 minutes. Add walnut paste and 1 cup of water, mix well. Turn heat down to medium low and simmer, partly covered for 20 minutes. Add pomegranate molasses, sautéed chicken, butternut squash, pumpkin puree, cinnamon, and 1 cup of water or chicken broth. Mix well.
Adjust flavors as needed with sugar, salt and pomegranate molasses to reach the desired balance of tangy and sweet that perfectly suits your taste. Gently simmer, covered, for approximately 45 minutes or until the sauce is a dark rich walnut color with a layer of oil on the surface from the walnuts. If the sauce is too thick or looks dry, add additional water or chicken broth.
Serve over rice or couscous and garnish with fresh pomegranate arils and chopped parsley.