Put carrot, onion, and celery in a heavy pan with the 1/4 cup butter, thyme, and bay leaf.
Simmer until the vegetables are tender.
Discard bay leaf and season with salt and pepper.
Wash shrimp.
Split shells down the back with scissors; remove sand veins.
Loosen shells, but do not remove.
Put unshelled shrimp in a saucepan.
Warm brandy, pour over shrimp, and set aflame.
When flame dies, add vegetables and wine.
Cover and cook 10 minutes.
Remove shrimp with slotted spoon to a rimmed platter; keep warm.
Beat egg yolks with a little of the hot sauce in a cup; then stir into the sauce.
Cook, stirring constantly, over very low heat just until thick (don't boil).
Pour over shrimp.