Search warm and high before you search far
A bearded dragon runs on sunlight, and that instinct is your search plan. Loose in a house, a dragon does not wander randomly: he makes for the brightest, warmest place he can find, basks until his body temperature drops, then squeezes into the nearest dark crevice to sit the cold hours out. Indoors that means windowsills and sun patches first, the warm tops of electronics second, and a still, flattened dragon in a cold corner third — and that third one is the one people miss.
Close the door of the room he vanished from, look at how the enclosure failed — a sliding door left a crack open, a screen top pushed up at one corner — and begin while the day is still warm enough for him to be moving.
Room by room, the way a dragon sees it
Start where the sun lands. Check windowsills behind the curtains, patches of sunlit carpet, and the tops of radiators and warm shelves, then work downward: behind the television and game consoles, on top of bookcases, and under sofas and beds at the edges where a dragon flattens himself into the shadow line. Dragons climb better than their stocky build suggests, using curtain fabric and upholstery as ladders, so scan vertical routes between floor and shelf, not just the surfaces themselves.
Feel with your eyes for stillness rather than movement. A cooled bearded dragon tucks his limbs in, darkens, and stays absolutely motionless, and in that state he reads as a shadow or a fold of fabric. Sweep each room's edges slowly with a flashlight even in daylight, because the beam separates a scaled outline from the clutter around it.
A basking station on the floor
Rather than only hunting him, invite him. Set his basking lamp safely over a flat tile or slate on the floor of the room he escaped in, with a shallow water dish and a small plate of his favorite greens — the foods he reliably runs to — beside it. A dragon whose body has cooled overnight will make for the first convincing patch of summer he finds, and a basking spot at floor level with food is exactly that. Check the station quietly every hour or so rather than hovering; dragons avoid crossing a room while large shapes are moving around in it.
Leave the emptied enclosure open on the floor nearby as well. Some dragons return to the one piece of territory that smells like themselves, especially toward evening when they look for a sleeping crevice.
The dragon who got outside
Outdoors the clock matters more, because a bearded dragon cannot regulate against a cold night and is exposed to cats and birds. Search the warmest microclimates first: sun-baked paving, the bases of walls that catch afternoon sun, rock piles, and the gaps under decking, moving outward from the door he most likely used. Tell neighbors the same day and ask them to check their own sun traps before evening.
Once he is home, warm him gradually under his lamp rather than in hot water, offer water, and watch for scrapes from squeezing through gaps. Then refit the failure honestly — a latched door, a screen top that cannot be pushed from inside. If your dragon does slip out and a neighbor finds him first, KinsLe's Lost Pet Alert System lets that finder message you directly through his profile, so he is back under his lamp hours sooner.