Heat a large oven-safe shallow pan over medium heat with a tablespoon of the oil. Once hot, add the dried chiles and cook until just beginning to soften, about 3 minutes. Add 1 cup hot water, turn off the heat and put on the lid. Let sit until the chiles are completely softened, about 5 minutes. Transfer to a blender and puree until smooth. Using a rubber spatula, push the mixture through a fine-mesh strainer into a small bowl. Set aside.
Wipe the skillet as clean as possible, and place over moderate heat with the remaining oil. Season the chicken with salt and pepper. Once the pan is hot, place the chicken in a single layer and brown on both sides, about 3 to 4 minutes per side. Transfer to a plate and set aside. Stir in the onion and garlic and cook until softened. Pour the chile sauce into the skillet and bring to a simmer. Then nestle the chicken back into the sauce and cook, slightly covered, over low heat, until cooked through and soft enough to shred with two forks, about 15 to 20 minutes.
To make the cheese sauce, set a medium saucepan over moderate heat with the butter. Once melted, stir in the flour and cook for about 30 seconds. Whisk in the milk and cook until it has thickened. Season with salt, pepper and the rest of the spices. Remove from the heat and stir 1 1/2 cups of each cheese (reserving 1/2 cup of each for later) until fully melted through. Keep the sauce warm until ready to use.
Preheat oven to 375 degrees F.
Bring a large pot of water to a boil and season liberally with salt. Cook the pasta according to package directions. Drain and return to the pot. Stir in the cheese sauce, hominy and the chicken and chile sauce until well-combined. Pour into a baking dish or the pan where you cooked the chicken. Top with the reserved cheese and bake until bubbly and golden brown, about 15 to 20 minutes. Remove from the oven and allow to cool slightly.
Serve in bowls and garnish with desired toppings. Enjoy!