I purchased a used cassette on ebay. The 12 tooth cog contained 11 teeth and what looks like a cutout where the 12 tooth would normally be. The cog is stamped 12t. Is this broken or is it engineered to be a 12 tooth cog with 11 teeth? Thanks.

Sounds like the SRAM OpenGlide design. What counts is the effective radius of the cog as a whole, so that's kosher insofar as it goes. But an OpenGlide-style cassette will have a tooth missing from each cog, not just the first cog.

+1 Don't worry about it, the cogs are the advertised size - circumference is more important than number of teeth.