Heat 2 tablespoons oil in a small pan over medium heat. Add walnuts and saute for 3-4 minutes, stirring frequently (don't burn it). Add the pomegranate paste and saute for 3 more minutes, stirring frequently. Set aside.
Heat 3 tablespoons oil in a pot over medium heat. Add onion and saute until tender (about 5-6 minutes). Increase the heat to medium-high; add chicken and saute until no longer pink.
Then add turmeric powder, water, salt and pepper. Cover the pot and bring to a boil. Reduce heat to medium-low and cook covered for 5 minutes. Then add the pomegranate-walnut mixture; combine well.
Cover the pot and simmer over medium-low heat until the stew is thickened and the natural oil is released from walnuts (about 2 hours or more), stirring occasionally (it is ok if chicken starts to fall apart). Add more COLD water during cooking if necessary.
Once the stew color is dark brown, add sugar and cook for 15 more minutes. Taste and adjust seasonings. Serve with Persian Rice.