The birria burger is defined by the consommé, not the patty. The braising liquid from beef birria, a deep red chile-and-spice broth thick with rendered fat, is carried into the burger in two places: the bun is dipped in it before griddling, and a cup of it comes alongside for dipping the assembled sandwich. That broth is the whole sandwich. It is what makes a birria burger a birria burger rather than a smash burger with a Mexican name, and every other decision in the build is arranged to keep the consommé in front.
The craft is in managing a wet sandwich on purpose. The beef is either formed into a patty from the chile-braised meat itself or run as a standard smash patty seared hard on a flat-top for crust, then crowned with Oaxaca cheese, the soft stringy melting cheese that slumps into the meat and glues the build the way American cheese does on a plain cheeseburger. The bun is the structural problem and the structural solution at once: dipped in the consommé fat so the surface takes on the chile flavor, then griddled so that fat caramelizes into a sealed, crisp face instead of soaking straight through to collapse. The dip on the side means the eater controls how far past the edge of structural failure to push, the same logic the French dip and the Italian beef run on, applied to ground beef and a chile broth. The fat in the consommé is doing flavor work and lubrication work both, and the sandwich is eaten fast while the bun still holds.
The variations track the birria family it borrows from. The quesabirria build leans harder into the cheese and the crisped tortilla idea; the all-meat version skips the ground patty entirely for shredded chile-braised beef; the chicken and goat readings swap the protein while keeping the consommé fixed. These belong to the wider American taco, burrito, and wrap map and to the broader argument about what a burger is once a region gets hold of it. Each of those deserves its own article rather than being crowded in here.