To make lobster stock: Split the lobster carcasses down the middle and remove the grain sac.
Cook the carrot,celery rib, garlic cloves, bay leaf, and thyme sprig in the olive oil in a 6- to 8-quart pot over moderate heat until softened.
Stir in the lobster carcasses and water. Bring to a boil and simmer until reduced to about 6 cups, about 11/2 hours.
Strain through a cheesecloth-lined fine sieve. Cool and refrigerate up to 4 days or freeze up to 3 months.