Heat the lard in a skillet, cook the garlic for 20 seconds, then add the beans, salt, and enough reserved liquid to mash them into a smooth spread. Keep warm.
Rub the steak with oil, salt, and pepper. Grill over high heat for 3 to 5 minutes per side. Rest for 10 minutes, then slice thinly across the grain.
Toast and Top:
Heat a large dry comal or griddle. Toast each tlayuda tortilla for 2 to 3 minutes per side until dry, lightly spotted, and crisp but still foldable.
Spread each tortilla with beans. Add quesillo and return to the griddle over low heat until the cheese softens.
Finish:
Top with cabbage, tomato, avocado, and grilled beef.
Serve open or fold into large wedges. Offer lime and salsa at the table.
Notes
True tlayuda tortillas are larger and drier than standard flour tortillas; use the largest thin corn tortillas available.
Keep toppings restrained so the crisp tortilla remains easy to lift or fold.