Crumble chorizo into a heavy, 12-inch skillet or Dutch oven. Cook over low heat to render the fat, but do not allow it to brown. Strain off fat, returning 3 tablespoons of fat to the skillet. Add the onion and garlic and cook over low heat, stirring constantly until onions are wilted and transparent. Add tomatoes, chilies, and adobo sauce. Increase heat to medium and cook, stirring often, until all moisture has been absorbed and the mixture is dry, about 15 minutes. Stir in shredded chicken and broth. Cook another 3 to 4 minutes, until chicken is heated through. Season to taste with salt and pepper.