Probably the "easiest" way to do this is to use the Ubuntu mainline kernel PPA, but download the .deb files individually. If you want to download kernels for 5.11.20 go here:
Kernel 5.11.20 from Ubuntu mainline PPA: https://kernel.ubuntu.com/~kernel-ppa/mainline/v5.11.22/amd64/
You'll find a bunch of .deb files there, the ones you need are these ...
I experienced the same issue, which seems to be caused by the grub.cfg file not being written to /boot/efi/EFI/ubuntu/grub/grub.cfg, just to /boot/grub/grub.cfg.
What I did was:
Created a backup of /boot/efi/EFI/ubuntu/grub/grub.cfg
Copied /boot/grub/grub.cfg to /boot/efi/EFI/ubuntu/grub/grub.cfg
After rebooting I was able to select newer kernels from the ...