The Yale blue burger is the New Haven build defined by a cheese chosen to do the opposite of what a burger cheese usually does. American and Swiss are picked to melt into a smooth, binding coat that glues the patty together; blue cheese is picked because it does not. It softens and slumps under the heat of the patty but keeps its crumbly structure and its pungent, salty, sharp punch, so it sits on the beef as distinct pockets rather than as a unifying film. The cheese is not the binder here; it is a loud, assertive counterweight to the beef fat, and that contrast, not melt, is the entire reason this burger has its own name.
The craft follows from a cheese that stays sharp instead of mellowing. The patty is cooked to drive a good seared crust, because a strong blue needs the depth of a well-browned beef surface under it or the cheese simply overwhelms a pale patty. The blue is added late and warmed by the patty rather than melted hard, so it goes soft and clinging but holds its bite; pushed too far it would lose the pungency that is the point of ordering it. The bun is soft and lightly toasted so it compresses to the patty and soaks fat without competing, the standard burger logic, and the garnish is kept restrained, because a sharp, salty cheese does not want a crowded build fighting it. Often the only additions are something cool and something acidic, set to cut the richness of beef fat and blue together rather than to add another loud flavor.
The variations stay inside the strong-cheese frame. A milder Gorgonzola or a sauced blue dressing softens the punch; adding bacon doubles down on salt and fat against the same cheese. It is one regional reading of the American burger, distinct from the smashed, the steamed, and the cheese-stuffed builds, which make their structural arguments elsewhere. Those deserve their own articles rather than being crowded in here.