Prefered methods would be to use some out of band management possibilities for this. For example Intel AMT or DRAC(this kind goes by many names, e.g kvm, lom,ilo)
If you dont have any out of band management possibilities, then the linux flashrom utility might be the easiest option, but as mentioned by myron-semack flashrom is certanly a risk since its unsupported by your bios provider.
Also, upgrading with flashrom might not even be possible. For example if your cpu has Intel ME running(More info here). In this case you need to use the bios maintainers utility, this is often only available under dos.
If your only fallback is to use the bios maintainers dos-tool then this method might be an option:
Create a (free)dos image with an autoexec.bat containing a call to the AMI update utility(there are no need to create a partition on the HDD)
I see two ways to booting into the dos image without physical access:
Pxe boot into the image. If pxe is not enabled and you dont have physical access this might not be possible though.
Change the bootloader to have a bootentry for the (free)dos image. See example of grub configuration for this here.
After modifying grub.cfg with the new entry, execute "grub-reboot X" where X is the grub id of your freedos bootoption.
I've recently upgraded a couple of machines by booting into dos with method "2" , without having physical access. I used the grub-memdisk packages to boot an image file under "/boot/images/" as described in the link provided above.
I highly encourage to try this locally before running this on a remote machine though, this is a very risky maneuver! (I certanly did)
Dont forget to add "reboot" after the call to the flash-upgrade utility in autoexec.bat. Also make sure the grub-default is your linux-partition offcourse.