"Is the cost basis that's reported on my 1099-B not the correct cost basis?"
I have no idea, of course, since I know nothing about the trade. The original poster mentioned "RSU" and "zero" cost basis and that was very clear.
"how should I determine the per share basis for each lot?"
The per share calculation is: (compensation created by the vesting) divided by (GROSS shares that vested)
I would be surprised if your employer didn't give that level of detail since you DO need it. You didn't say that the basis reported by the broker was $0 so if they have put some number in the basis box they might have reported to you with the correct basis.
You can always use the stock's closing price on the vesting date as a pretty close approximation of what the employer used as the per share "fair market value". So if the 1099-B does have a figure for the basis and you know the number of shares sold from each lot and each lot's vesting date you can test that out yourself.
