A brioche burger is defined by the bun, not the patty, because changing the bun changes everything the sandwich is asking the patty to do. Brioche is an enriched dough: egg and butter worked through it give it a tight, tender, faintly sweet crumb and a soft, slightly resinous crust that browns deep gold. Put a seared beef patty inside that instead of a plain bun and the whole reading shifts. The sweetness reads against the salt and char of the meat the way a relish would, the soft crumb compresses to the patty rather than fighting it, and the close grain holds together under load where an open white bun would tear. The patty can be excellent or ordinary; the brioche is the decision that makes it a brioche burger.
The craft is structural, and it is mostly about water. Brioche is tender precisely because it is rich, and a tender bun put under a hot, juicy patty and a wet sauce will go to paste fast, so the inner faces are toasted on the flat-top until they form a thin sealed crust that resists the juice and adds a contrast the soft crumb otherwise lacks. The bun is sized to the patty so the bread-to-beef ratio stays honest, since brioche has a flavour of its own and too much of it turns the sandwich into sweet bread with meat in it. The crumb's tight structure is doing real work: it takes the compression of the bite without collapsing, holds the patty and its juices in a sealed pocket, and lets the whole thing be eaten in the hand rather than disintegrating halfway.
The variations stay on the enriched-bun shelf and the premium-burger logic that put the patty there. A potato bun trades brioche's butter-sweet note for a denser, more savoury softness; a milk bun is softer and plainer again; a glazed and seeded brioche pushes the colour and the crust further. The patty itself forks into the dry-aged build, the smashed style with its lace-edged crust, and the stacked double, each a different sandwich on the same sweet bread. Each of those deserves its own article rather than being crowded in here.