Focaccia con le cipolle is the Ligurian dimpled focaccia handed over to the onion. The base is the familiar one: an olive-oil-rich dough pressed thin into a tray, knuckled into deep dimples, and brushed with a brine of salty water before baking so the wells stay moist. What sets this version is the topping. Thinly sliced onions, sweet yellow or white, are spread across the surface before the disc goes into the oven. As it bakes the onion softens, slumps into the oiled dimples, and the slices at the edges of each ridge catch and brown into something sweet and faintly bitter. The result reads less like a bread with onions on it and more like a single thing, the allium and the oiled crumb fused under heat.
The craft is the management of moisture. Raw onion sheds a lot of water as it cooks, and dumped on too thickly it steams the dough into something pale and slack. The Ligurian fix is to slice the onion fine and lay it in a single, even layer so it caramelizes rather than stewing, and to keep the oil and salt-brine generous enough that the crumb stays rich underneath while the surface crisps. The sweetness the onion develops is the whole reason for the variant: it plays against the salt of the crust and the bitterness of good olive oil, and a con le cipolle baked timidly, with under-browned onion, loses that contrast and tastes only of bread. Split horizontally it carries cured meat well, the sweet onion doing the work a relish would, but in Liguria it is most often eaten plain and warm, by the cut, off the tray.
The named relatives change the topping rather than the method. The focaccia con le olive presses olives into the same dimpled base for a briny rather than sweet result. The plain Genovese round is the unadorned parent of all of them, and the thinner, crisper town styles along the coast take the same dough in different directions. Each of those is its own subject and deserves its own article rather than being crowded in here.