How can I pick up the blockchain download where it left off on another computer using an EHD?
Even if you transfer all the data from a fully synced client, the bitcoin client still needs to validate those blocks. It will still take a while, but should be quicker than downloading all the data and verifying.     

ANSWER IS NOT ACCURATE.
EDIT: If you copy the chainstate, there shouldn't be any work the receiving system should perform anymore - the software wouldn't even know it was moved. It's possible the wrong directory was used.