1. Preheat broiler.
2. Place pepper halves, skin side up, on a foil-lined baking sheet. Broil 6 minutes or until blackened. Place in a paper bag, and fold to close tightly. Let stand for 15 minutes. Peel and chop; set aside.
3. Heat oil in a large Dutch oven coated with cooking spray over medium heat. Add onion to pan; cook 6 minutes, stirring occasionally. Add garlic; cook 1 minute, stirring occasionally. Add chile powder and oregano; cook 1 minute, stirring constantly. Stir in 4 cups water, broth, and hominy; bring to a boil. Reduce heat, and simmer, uncovered, 10 minutes. Stir in Anaheim chiles and turkey; cook for 2 minutes. Stir in cilantro and salt; cook 3 minutes. Ladle about 1 1/3 cups soup into each of 8 bowls. Top each serving with 1 tablespoon pumpkinseed kernels, 1 tablespoon radishes, 1 tablespoon green onions, and 1 tablespoon cheese. Serve with lime wedges, if desired.
Beer note: With Anaheim and guajillo chiles lending their subtle heat, reach for a flavorful chilled beer, like a Scottish-style ale. Oskar Blues Old Chub Scotch Ale ($8.99/six-pack), from Colorado, has a rich, malty sweetness, hinting of caramel, that works to balance the peppery posole, while the beer's dark chocolate, toasted nut, and smoky notes complement the roasted pumpkinseeds in this richly layered soup. --Jeffery Lindenmuth