Preheat oven to 400°. Remove stem and seeds from chile. Combine chile and boiling water in a small bowl; cover and let stand 1 hour. Place chile and water in a food processor, and process until smooth. Spoon chile purée into a small bowl; set aside.
Combine squash, 2 teaspoons oil, 1/8 teaspoon salt, and 1/8 teaspoon pepper in a 13 x 9-inch baking dish coated with cooking spray; toss well. Cover and bake at 400° for 5 minutes.
Sprinkle 1/4 teaspoon salt and 1/4 teaspoon pepper over chicken. Heat 2 teaspoons oil in a large Dutch oven over medium heat. Add chicken and mushrooms, and sauté 5 minutes. Remove chicken mixture from pan; set aside. Heat remaining 2 teaspoons oil in pan over medium-high heat. Add onion, bell peppers, and garlic; sauté 3 minutes. Add 1 tablespoon chile purée, squash, chicken mixture, remaining 1/8 teaspoon salt, remaining 1/8 teaspoon pepper, tomato, zucchini, cinnamon, cumin, and hominy; cover, reduce heat, and simmer 25 minutes, stirring occasionally. Remove from heat; stir in yogurt. Ladle stew into individual bowls; sprinkle with cilantro. Serve with additional chile purée, if desired.
Note: Chef Heidel prefers to serve this stew in a small, hollowed-out sourdough round.