Take the Picard leek tart, cut a wedge, and eat it in the hand: that is the whole move here. The flamiche is a savory tart of leeks softened slowly in butter until they collapse into something sweet and silky, bound with cream and egg, set in a pastry case and baked. Eaten sandwich-style means a wedge in the hand or a piece folded into bread, and the leek filling is the entire proposition: a vegetable cooked down past the point of being a vegetable, mellow, buttery, and faintly sweet, carried by its own pastry rather than by a baguette.
The character of cooked leek sets the design. Slow-stewed leek loses its bite entirely and turns soft and gentle, with a sweetness closer to onion confit than to anything sharp, and the cream-and-egg bind sets it into a sliceable custard rather than leaving it loose. That is why the flamiche holds together as a hand food at all: the bind is structural, and a cold or barely-warm wedge keeps its shape where a hot one would slump. The pastry is doing the job bread does elsewhere, a firm dry edge against a soft rich interior, which is also why nothing is added inside; the leek is already sweet, already rich, already seasoned, and a competing filling would only blur it. It eats best slightly warm, never hot enough to go runny, never so cold the butter sets hard.
Variations are mostly the bind and the dairy. Some versions lean harder on cream for a looser, richer interior; some carry a little Maroilles or another regional cheese worked into the leek for a sharper, washed-rind edge against the sweetness; some keep it austere, leek and butter and not much else. The frame holds across all of them: stewed leek, a setting bind, a pastry case, eaten warm by hand. The Sandwich à la Flamiche belongs with the dishes-folded-into-bread the catalog groups under Plat-en-Sandwich, the French tradition of carrying a cooked dish in the form of a sandwich. Its particular contribution is a vegetable cooked into a sweet, custard-set filling that needs no bread because its own pastry already plays that part.