You should use a depletion type organic FET with slightly positive threshold voltage and very small subthreshold swing. However, in order to answer your question properly, more details regarding your experiment and device operation requirements are needed.

I agree with Leszek, more information on your setup would be nice. Finding anything with a 10mV gate voltage will be tricky though, at least if you are looking for an OFET. I think just about any reasonably well-working OFET these days can probably do the 10kHz, but you need to be more specific as to output current density, desired device area etc. Do you want to work in linear or saturation mode and what exactly is the FET supposed to do?