The Brioche Perdue Garnie is what happens when yesterday's brioche refuses to be thrown out. Stale or slightly-dry brioche is dipped in a custard of egg, milk, sugar, and a little vanilla, then pan-fried in butter until the outside crisps to gold and the inside softens into something between bread and pudding. Two slices of this pain perdu, sandwich-style, hold a filling. The savory versions land thin-sliced ham and a slice of cheese between the two custarded pieces; the sweeter end of the spectrum reaches for crème de marrons, fig jam, or even a thin layer of foie gras. The result is a sandwich that sits between brunch, dessert, and supper.
The technique is older than the sandwich. Pain perdu, literally "lost bread," is the classical French solution to stale bread, going back to medieval cookery as a way to rescue what would otherwise be wasted. Using brioche rather than baguette is a richer-end-of-town move, since brioche's high egg and butter content makes the custard dip both unnecessary and gloriously redundant. The custarded brioche behaves differently from custarded baguette: it stays softer, takes color faster, and absorbs filling moisture in a way that makes the sandwich more cohesive than it has any right to be. The right pan temperature, in the range of medium rather than hot, is what keeps the bread from charring before the custard has set.
The variations split along the savory-sweet line. The savory Brioche Perdue with ham and Comté is the bistro version, served warm, often as a small plate at a wine bar. The Brioche Perdue with béchamel and ham slides into Croque-Monsieur territory and is the late-night menu item at certain Lyonnais bouchons. The sweet end stays closer to the dessert tradition: brioche perdue with confiture de lait, with sautéed apples and Calvados, with mascarpone and berries. The Brioche Perdue au Foie Gras is the holiday version that shows up at New Year's. The broader Brioche Garnie family covers the wider universe of enriched-bread French sandwiches, and the pain-perdu version is the most architecturally adventurous corner of that small map. Whether it counts as a sandwich at all is the kind of question the French food world tends to answer with another glass of wine.