Our best guess right now is that it's the periostracum (outer covering) from a large bivalve. (It measured ~6.5 cm long by ~5 cm wide.)
However, the material and its transparency seem different than other bivalve coverings that we've seen, so we're open to any other thoughts or suggestions. The inner surface appeared to have a chalky white residue (perhaps from the middle shell layer of a clam?).
And even if it is the outer covering of a clam, we are uncertain about which species it's from.
Here's another view, this time from the side:
Let me know if you have any ideas about its identity. I'm still working on it, but it would great to solve this mystery!
ADDENDUM (28 November 2016): Several people have suggested this could be the operculum ("trap-door") of a large snail, e.g., a whelk. That's a strong possibility, and at this time I think I'm leaning that way.
ADDENDUM (1 December 2016): For the identity of this mystery object, see the post on 28 November 2016.