Have I toasted my hashing boards?
Maybe, but unlikely, the miner has temp sensors that stop it from mining when temps go past a certain degree, if you have not disabled that then it's unlikely that you toasted them.
Is there a way to tell beyond talking to the control board?
Yes you could measure the voltage and resistance of each individual chip using a multimeter, or buy a special fixture tool, but your problem seems like a bad control board, I believe that if you Sdcard the miner using recovery files and then flash 2017-2018 firmware your miner might just come back to life.
Are they repairable with a soldering rework station?
If it's a bad heatsink or two, yes, you just need the adhesive and a heat gun, but this isn't a common fault with the S9s, usually, when an S9 hashboard is dead it's so because of a toasted chip or a few of them, which means you will need to order some chips and obviously a fixture tool which doesn't make much sense since this is just a single S9.
