I search a off the shelf RF feedthru that build from Teflon and Centeral metal pin only to connect to 2 stack of RF boards, i will prepare a hole in the chasis and insert the RF feedthru to it and solder the 2 boards toghether.
the impedance of the hole+teflon+pin will be 50 ohm.
Make sure the "equivalent ground return" part is satisfied, or you will be dissapointed. A teflon sleave does nothing to connect the 2 board grounds, and trying to solder two boards together is not as easy as it sounds...solder voids are common.
The design geometry should be clarified. If both PCB boards are sitting with their ground plane on both sides of the metal plate, there would be a sufficient ground return.
I do it in the past without TEFLON at frequency 2GHz and it is work well. now the frequency is 5GHz so i add the teflon in order to concentrate the pin.
Search the web using "RF feedthrough" for some possibilities.
Pins and dielectric, often ceramics for hermetic seals on rf feedthroughs are common. You could use a short length of coax or if you do not want the shield just pull the dielectric and center conductor out of the cable. Look at the mounting rf interfaces for "field replaceable connectors for ideas and possibly just the part you need.
I would pay close attention to the transition from what-ever transmission line you have on the PCB's and the feedthrough. If you do not use care there may be a significant discontinuity (high s11, low s21).
It looks like what you are trying to accomplish is to create what is known as a printed circuit "via". It essentially is a hole in the board that is then plated to produce the conductor.
There is a easy way I used to do. Just find a RF cable, remove the outside cover, then remove shielding ---outside GND, just keep dielectric and inside conductor. Made a hole in the metal, put it in the hole, then metal can be the outside conductor as a RF cable. The hole diameter should be about 0.2mm larger than the dielectric diameter. This way can work up to 8GHz if you don't care about the power. Maybe several dB loss.