Manoushe b'Bayd (منقوشة بالبيض) is the Lebanese flatbread with egg cracked onto it, baked so the egg sets into the dough. The angle is timing the egg against the bread. The egg can go on raw before the round enters the oven, so it bakes from raw to set as the crust does, or it can be added partway through so the white firms while the yolk stays loose. Either way the defining trait is that the egg cooks on the bread, not in a separate pan and then laid over.
The build is the standard manoushe method with egg as the dressing, often over a base layer. The dough is portioned, relaxed, then hand-pressed and dimpled into a round, the dimples now doing real work by giving the egg shallow wells so it does not all run off the edge. A base is common underneath, za'atar, cheese such as akkawi, or awarma, both for flavor and to give the egg something to grip. One or more eggs are cracked on, sometimes left whole for a set yolk, sometimes beaten and spread for an even cooked layer. It bakes in a wood-fired oven or on a saj until the base is cooked and the egg is set to the desired point. It is eaten flat or folded warm, often with tomato, onion, or mint added off the oven. Good execution controls the egg so the white is fully cooked while the yolk stays where it is wanted, keeps the round level so the egg does not pool to one side, and times it so the base bakes through without the egg going hard. Sloppy execution lets the egg run off a too-flat round, overbakes the yolk into a chalky disc, or underbakes the base so the dough is raw under a cooked egg.
It shifts mostly by the base under the egg and by whether the yolk is set or runny. Over za'atar it is herbal and lean; over cheese it is rich and stretchy; over awarma it is the most decadent, egg and preserved lamb together. A runny yolk makes it a sauce that coats the bread when folded; a fully set yolk makes it sturdier and more portable. It sits squarely in the manoushe family beside the za'atar baseline, the akkawi cheese version, the awarma version, and the broad breakfast reading, and it is the egg expression of that family. What manoushe b'bayd reliably delivers is egg cooked directly onto stretched dough, usually over a savory base, eaten warm and folded.