Divide the beef into 4 portions. Form the beef into loosely packed rounds — like a meatball. Set aside while you prep the grill.
Heat a grill to two-zone cooking. Toast the burger buns over high heat and then set aside.
Heat a large griddle or skillet over direct heat. Add four mounds of Oaxaca cheese and as it begins to melt, top each with the birria. Top with about 2 tablespoons of consomme and cook until the cheese is melted and crispy on the bottom. Remove from the pan and keep warm while you cook the burger patties.
Working in batches of two, add two beef rounds to the skillet at a time – season each with salt. Let the burger balls cook until some fat pools at the bottom of the burger. Use a large flat spatula to press down hard on each patty for 2 minutes. The burger should be super thin and have a diameter that is at least two inches wider than the bun (if your pan is too small you may only be able to do one at a time).
When the juices in the middle of the burger start to bubble up, flip the burger. Cook another 2 to 3 minutes then top each patty with Oaxaca cheese and consomme, cook until melted.
Build the burgers by topping the bottom bun with a burger patty, then the cheese skirted birria, a few chips and some salsa verde before topping with the top bun.