The Tartine Gratinée is the savory open-face slice taken to heat: cheese on a single slice of bread, run under a broiler until it melts and browns, eaten warm as a quick lunch at a counter. The defining element is the gratin, the lacquered, blistered top the heat builds, and that finish is the whole difference between this and the Tartine au Fromage, which serves the same cheese cold. This is a midday slice and a bistro slice, something a kitchen can put under the salamander and send out fast, not a breakfast or an apéro item. The region recorded for it is Paris.
The craft is what heat does to a single slice carrying melted cheese, and the whole build is arranged around that. The cheese has to be a melting one, a Gruyère or a Comté or an Emmental, grated or sliced so it flows evenly and lacquers rather than splitting into oil and string. The bread is the load-bearing problem: a single slice has no second slice bracing it, so it needs enough crust and body that it crisps under the broiler instead of going limp under the weight of molten cheese, which is why a firm cut of country bread serves better than anything soft. The top blisters and takes color while the crumb just beneath stays soft and the crust at the edge goes hard, three states on one slice at once. It comes off the heat ready and goes downhill fast: this is best within a minute or two, while the cheese is still molten and the crust has not yet softened back under the steam from inside.
The variations are the rest of this family, each a single change for its own moment. Serve the cheese cold and unmelted and you have the Tartine au Fromage, the room-temperature version; lay ham and butter along a baguette and it is the café Tartine Parisienne. A slice of ham slipped under the cheese before it goes to the heat is the common addition, and the slice tilts toward its grilled relatives from there. Strip it to the bread alone and it is the plain Tartine Beurrée. The Tartine Gratinée sits in the open-face tradition the catalog groups under Tartine, and its particular contribution is the broiled, lacquered top that a single sturdy slice exists to hold up.