Daniel,
This is not possible with the counters on an E-Series device because they do not support a start trigger (DAQ-STC chip). The NI 660x counter devices have the NI-TIO chip, and they do support a start trigger. So, you could connect your E-Series device and an NI 660x with a RTSI cable and route the start trigger from the E-Series device to the NI 660x. If you do not have access to an NI 660x, below is a link to the product page:
Counter/Timer DevicesGood luck with your application.
Spencer S.