https://github.com/wkoszek/freebsd_netfpga
https://wiki.freebsd.org/FPGA/
https://www.freshports.org/devel/xc3sprog/
It's a FreeBSD NetFPGA driver for an older 1G card that I wrote during my internship 8 years ago, and a little ecosystem to make development more functional. I could program the FPGA from the FreeBSD and synthesize the code on FreeBSD that time.
What you say is very hard, though. Speeds which are achieved with modern ASICs are hard to compete with, because ASICs do a lot of advanced stuff with DMAs, interrupts, checksum offloading. This might be doable with an expensive hardware, but nobody in DYI community has money to do this.
Additionally the tools for synthesis are proprietary, and everything touching FPGA is pretty much proprietary too. Looking into date/author of the synthesis is as far as I could get:
https://github.com/wkoszek/libxbf
(can only open file and tell you where the bitstream starts; not what it actually is) So the road to complete freedom goes through ASIC world, in my opinion.