Rinse the beans and place in a large pot or casserole, cover with 14 cups of water, add the apple cider vinegar and bring to a rolling boil. Cover, reduce the heat to low and simmer for 1 hour to 1 hour and 15 minutes, until they are soft. Add one tablespoon of salt, stir and cook for another 5 to 10 minutes. Set aside. (If using canned beans, skip this step.)

Heat a small saucepan over medium heat. Once hot add the chiles and toast for about a minute per side, letting them begin to crisp up and change and darken color, then cover with water. Bring to a boil, reduce to a simmer, and cook for 8 to 9 minutes until rehydrated and plumped up. Set aside.

Heat the oil in a large pot or casserole, over medium heat. Once hot, add the onion and cook for 2 minutes, just until it begins to soften. Add the garlic, stir and cook until fragrant, less than a minute. Add the potatoes and carrots, stir, and cook for 7 to 8 minutes, until they have softened. Incorporate tomatoes, add 1 teaspoon salt, stir and cook for 3 to 4 minutes. Incorporate the beans and all their broth (if using canned, add water or broth as needed to make it soupy), let it come to a simmer and cook for 10 to 15 minutes. Set aside.

Working in batches if need be, puree the cooked bean mixture in a blender, along with the rehydrated chiles and ½ cup of their cooking liquid.

Heat the lard in a large pot or casserole. Once melted and hot, incorporate all the pureed beans. Cover with a lid ajar, as the beans will splatter and jump, and cook for 10 to 12 minutes, stirring regularly so they don't create a crust on the bottom of the pan that can burn, until they have the consistency of a potato puree.

Serve spread over tostadas, topped with avocado slices and Cotija cheese.