Heat a large skillet or griddle. Divide the birria into 3 inch round on the skillet, you may need to work in batches. Pour a little consome on to the beef to help reheat and steam the birria.
Top each round with Oaxaca cheese and cook until melted.
Dip tortillas in the consome and top the birria mounds with a tortilla each before flipping. Cook until the tortilla begins to crisp, then top each taco with onions and cilantro and fold to close.