Bacon and brie is the bacon roll that left the building site and walked into a gastropub, and the defining move is the cheese reading rather than a sauce. Brie is a soft, bloomy-rind cheese with a high fat content and a mild, mushroomy, faintly tangy flavour that turns liquid and slack under low heat. Putting it against salt-cured bacon swaps the usual brown-or-red acid counter for a fat-on-fat one: instead of vinegar cutting the bacon, the brie blankets it, the cheese's mildness absorbing the salt and the cure into something rounder and richer. This is not a thrift sandwich pretending otherwise. It is the bacon butty rebuilt as an indulgence, and the cheese, not a condiment, is the decision that makes it a different sandwich with its own name.
The build is a melt-and-balance problem. Brie has to be warmed enough to slump and coat without weeping all its fat into the bread, so it goes straight onto hot bacon fresh off the heat and lets residual warmth do the melting rather than a long grill that would split the cheese and soak the crumb. The bacon is cooked until the fat has rendered and the edge has crisped, which matters more here than in a plain butty: brie is rich and soft, and without a salty, firm, slightly crisp counter the whole sandwich reads as one heavy, fatty note with nothing to break it. The bread is usually a sturdier carrier than a standard soft roll, a ciabatta, a baguette, or a firm white bloomer, because molten high-fat cheese plus rendered bacon fat is a lot of grease and a tender bap would go to paste under it. A sharp accent often enters to do the job the cheese gave up, cranberry, a fruit chutney, or rocket, supplying the acid or pepper that cuts the richness the brie smoothed over.
The variations are the gastro reworkings of the same idea and the breakfast roll it descends from. Camembert deepens the cheese; a fruit element, cranberry or chutney, is a common fixed partner; rocket or spinach turns it toward a lunch sandwich. The plainer bacon rolls it grew out of, the brown-sauce and red-sauce camps and the egg builds, sit alongside it as the working version of the same starting point. Each of those deserves its own article rather than being crowded in here.