To make the refried beans: The day before, sort and rinse the beans to remove all impurities and broken or brittle beans. Place the beans and 3 cups water in a pot and cook over medium heat, covered with a lid, for 45 minutes, cooking the beans until they are soft enough to be easily mashed with the back of a spoon. To test, remove one bean and press it with the back of a spoon, if the bean presses into a smooth paste the beans are done. During the cooking, add hot water as needed to keep the beans from drying out. Leave enough cooking liquid to cover.
In a medium saucepan set to medium heat, fry the garlic and onions in the asiento until translucent, then add the beans, and puree constantly with an immersion blender. After 5 minutes, add the salt and cayenne pepper, continuing to stir with the immersion blender until you reach a smooth and slightly runny consistency. Adjust seasoning and set aside, covering the saucepan to keep the beans warm and moist.
To make the salsa: Roast the hatch peppers over an open flame, evenly blackening the skin and taking care not to burn the flesh. Remove the heads from the fresh chiles de árbol and toast in a dry pan over high heat for 3 minute along with the tomatillos, onion, and garlic, stirring constantly with a wooden spoon, taking care not to burn the chiles. Move all ingredients to a blender (including 3/4 cups water), blend until smooth, and season with salt at the end to taste. (If you want less heat, remove the seeds and membrane before toasting.)
To make the tlayuda: Remove the pork chorizo from the refrigerator 30 minutes prior to cooking or until it’s at room temperature. Add the olive oil to a small saucepan and cook the chorizo for about 5 minutes, stirring constantly. After cooking, strain out the chorizo oil and reserve. There should be about one tablespoon left over.
If your range doesn’t have a griddle, use a flat-top attachment or a large electric skillet on low heat to warm your tlayuda tortilla, but only begin this process once all ingredients are ready to go. Meanwhile, warm the beans in a saucepan over low heat. Using a spatula, apply a thin layer of chorizo oil to cover the tlayuda while it’s on the griddle. Add 2 tablespoons of refried beans, leaving a thin strip of the tortilla’s circumference exposed.
Add the Oaxaca cheese to a plancha over medium-low heat and over a drizzle of vegetable oil for about 1 to 2 minutes, until the top later melts and the plancha side gets a crispy skin. Quickly peel off the melted cheese with a spackling knife and place it melted side down on the tlayuda.
At the same time, fry the egg in oil briefly to achieve a runny yolk, then add salt and pepper to taste, and remove from the heat.
Add the crumbled queso fresco over the mini tlayuda, and place the tomato and avocado slices on top. Spoon over the chorizo, add the arugula, and place the fried egg on top. This should all be done quickly, monitoring the tlayuda so it doesn’t burn. You can remove it from the heat and place in on a cutting board to do some assembly if it gets too hot, but the object is to layer the tlayuda rapidly. Serve with salsa on the side.