It sounds like the shell was picked out of some twigs. I don't think a shell that is small enough for a pigeon to carry would be big enough to contain twigs. So he shell was 'caught' in the twigs and the pigeon 'freed' it.

"a half-shell from which a pigeon had removed all twigs one by one" "picked free of twigs" and "cleaned free of twigs" are the same except that "picked" is one by one. It is one by one because a bird could only remove one twig at a time.

It comes from the book "The Forbidden Wonder Of Birds And Nests" By Helen Macdonald. In which she discusses the moral implications of collecting birds eggs taken from birds nests, even if only taking one egg from a nest. White eggs are prized because they are not common 95% of bird eggs have colour or colour imperfections on a white egg. One day she came across half a white egg shell dropped on a lawn by a pigeon that was free of any twigs. Despite the white half egg shell being of a collectible pure white colour, she could not bring herself to take it home. It came out of the nest a nest clean with no twigs from the tree or nest.

To me it means that the pigeon brought half-shell and no twigs with it. I imagine the half-shell was clean without other debris. No twigs were attached to it.

We need more context to fully answer your question first off we need to know what is a half-shell (hyphenated) and combined to become one noun?. Let us suppose it is a half-shell that a bird would eat and like which is served in restaurants. The one half of a shell like a clam or oyster after it has been broken by the bird or chef to be eaten. In this scenario a bird having broken a shell into a half-shell. would presumably pick it clean of any twig. And twigs and twiglets doesn't automatically imply branches and twigs from a tree. A twig in biology such as in a half- shell of a sea creature also means the small nerves and blood vessels leaving the heart. or blood vessels. Twig any small shoot, blood vessel, nerve etc growing out from a main x. (not only a tree branch)