I'm having a similar problem with my ASRock Z77 Extreme4 board + GeForce GTX 1060 3GB + Fresco Logic FL1100 USB controller.

If I am passing through my 1060 + Fresco Logic, it will work from a cold boot, but it cannot handle 1 or 2 (if I'm lucky) shutdown/startup sequences before it crashes ESXi completely where the host stops pinging.

Interestingly, what I have found is that passing through the board's ASMedia USB controller (integrated, controls 2 USB ports on the back, rest is Intel USB) instead works just fine!

I haven't found any way to fix this. Tried many different combinations of advanced settings. So, I can't recommend this card..find something else! Onboard USB 3.0 passthrough works fine.