I don't think it seems incomplete or ambiguous. To me, it is pretty clear that it is turning the verb form of "done" - that is, not the adjective meaning "finis

I don't think it seems incomplete or ambiguous. To me, it is pretty clear that it is turning the verb form of "done" - that is, not the adjective meaning "finished", but the past tense of the verb "do" as in "fishing is an activity that people have done for a long time" - into an adjective modifying "thing". So a "done thing" is just "a thing that is being done". Using that fishing example again, it shifts from "fishing is a thing that is done by people" to just "fishing is a done thing". This "adjectivizing" technique is a common way that colloquialisms are created. (Maybe it's a bit weirder in this case because an adjective form of the word already exists, but has a different meaning.)