Intel thoughtfully has some ISO images of their BIOS flash upgrades, so you don’t need to worry about finding the right flash software for your operating system and then timidly hoping that all works OK. You burn the image to a CD and reboot, then it flashes for you (using a FreeDOS/ISOLINUX system).
However, if you have a SATA CD-ROM drive, the device driver in FreeDOS doesn’t support that. There is a SATA-compatible FreeDOS driver, but rather than rebuild Intel’s ISO, there’s an easier solution – make the BIOS emulate an IDE drive.
Go into BIOS Setup (F2 at boot), then Advanced … Drive Configuration, and set ‘Configure SATA as’ to ‘IDE’ (mine was AHCI) and ATA/IDE Mode to ‘Legacy’.
Reboot, allow the flash to succeed, then switch your BIOS settings back.
There’s nothing wrong with this method, but Intel should highlight it on their download page.