Pat the chicken dry with paper towels and season with salt and pepper.
Add 1-2 tablespoons of olive oil to a large Dutch oven set over medium-high heat. Working in batches, lightly brown the outside of the chicken thighs and drumsticks until they are golden brown. Remove to a clean plate and repeat, adding additional olive oil as needed.
Add the walnuts to the bowl of a large food processor. Process until the walnuts form a thick and chunky paste (you may need to scrape the sides of the food processor a couple of times to help move things along). As the walnuts are being processed, add a few tablespoons of cold water through the feed chute. Continue processing until the walnuts become a smooth beige-colored paste.
In the same Dutch oven, heat 1 tablespoon of olive oil over medium heat. Add the onions and garlic and cook for 1-2 minutes, stirring often. Add the walnut paste plus 1 cup of water. Mis well to combine. Reduce heat to low and simmer, partly covered, for 20 minutes.
Mix in the pomegranate molasses and add the browned chicken pieces, butternut squash, pumpkin puree, ground cinnamon, orange peel, and 1 cup of water (or chicken broth). Mix well.
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.
Adjust the flavors as needed with added sugar, salt, and/or pomegranate molasses to reach your preferred balance of tangy and sweet.
Serve over rice or couscous and garnish with fresh pomegranate arils, chopped parsley, or fresh thyme.