Remove seeds from 2 dried ancho or pasilla chiles; toast in a dry small skillet over medium-high heat, turning, until darkened and fragrant, about 4 minutes. Place in a medium bowl.
Add 2 cups hot water; let sit until softened, about 5 minutes.
Pulse chiles in a food processor with ½ cup soaking liquid, 1 garlic clove, and 2 tablespoons tomato paste until smooth.
Heat 2 tablespoons olive oil in a large pot over medium heat.
Cook 1 thinly sliced medium onion, stirring occasionally, until translucent, 6–8 minutes.
Add 2 tablespoons chile paste and cook, stirring, until thick and darkened, about 4 minutes.
Add two 15-ounce cans white hominy, rinsed, 8 cups turkey stock or low-sodium chicken broth, and 2 cups shredded cooked turkey meat; season with salt and pepper.
Simmer until flavors meld, 10–15 minutes.
Serve with tortilla chips, sliced avocado, cilantro sprigs, and lime wedges.
