GCC has a "bootstrap" build process. So when you try to build gcc-11 with only gcc-8 installed, it will build a temporary "stage 1" version of gcc-11 using gcc-8, then compile gcc-11 again using gcc-11-stage1. Thus no matter what you start with, the version of gcc-11 that comes out of the build process was effectively compiled with itself.

So all that matters is that gcc-8, or whatever "system compiler" was previously installed, is able to build a stage1 version of gcc-11 that runs well enough to compile the stage 2 version. It doesn't matter whether your system compiler is good at optimizing, and gcc's source code is deliberately written to use a fairly minimal set of language feature (at least for stage 1), so you are not likely to run into trouble with your system compiler having missing or buggy support for obscure corners of the language. Historically, the "system compiler" was often not gcc at all, but some compiler provided by the computer vendor or an unrelated third party, and so one couldn't rely much on its quality; gcc was designed with that in mind.

Theoretically your system compiler could have a bug which miscompiles gcc-11-stage1 in such a way that it appears to work, but itself miscompiles stage2. This is unlikely, and it's even less likely that it would happen in a way that wasn't obvious (e.g. the stage2 compiler simply segfaulting). If worried, there's an option to have stage2 build a stage3 compiler, and then check that both versions are identical. So as long as the build completes, you can be pretty confident that the final installed compiler is fine and unaffected by bugs in the original system compiler. (All that said, a reference to Ken Thompson's "Reflections on Trusting Trust" is obligatory here.)

So in practice, you don't need to worry about the version of gcc used to build a new version. Whatever you happen to have installed already, within reason, will be fine.

NOTE: The 6.5 Kernel reports being built as x86_64-linux-gnu-gcc-12 (Ubuntu 12.3.0-1ubuntu1~22.04) and this should be OK. This is because the kernel is not built on your system during install. See https://askubuntu.com/a/1171939/231142 for more info. During any builds it might give you a warning about not matching the installed version since they are not named the same. Installed version reports as gcc-12 (Ubuntu 12.3.0-1ubuntu1~22.04)

gcc-12 should already be installed, but if it isn't, you can install it or reinstall it by running:

sudo apt install --reinstall gcc-12

Then all you should have to do is update the link for gcc to go to the gcc-12 binary:

sudo ln -s -f /usr/bin/gcc-12 /usr/bin/gcc

You can set it back to 11 as well by doing the following:

sudo ln -s -f /usr/bin/gcc-11 /usr/bin/gcc

You can check it by running gcc --version.

terrance@terrance-ubuntu:~$ gcc --version
gcc (Ubuntu 12.3.0-1ubuntu1~22.04) 12.3.0
Copyright (C) 2022 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

This is not recommended but you can install version 11.3 by downloading gcc-11 (version 11.3) from here. And downloading the dependencies gcc-11-base and libgcc-11-dev.

Then, run sudo apt install /path/to/file where /path/to/file is the path to the actual downloaded file. If it can't find the file, you may need to put it in double quotes like "/path/to/file".

You will need to install the gcc-11-base package first, then libgcc-11-dev and finally, gcc-11.

For example, if it's downloaded to your Downloads directory:

sudo apt install $HOME/Downloads/gcc-11-base_11.3.0-6ubuntu1_amd64.deb

and

sudo apt install $HOME/Downloads/libgcc-11-dev_11.3.0-6ubuntu1_amd64.deb

and

sudo apt install $HOME/Downloads/gcc-11_11.3.0-6ubuntu1_amd64.deb

I'm pretty sure gcc only requires gcc-11 >11.2 so this should downgrade your version of gcc and when you're done compiling or whatever you need it for, simply run:

sudo apt update
sudo apt install --reinstall gcc-11 gcc-11-base libgcc-11-dev

to upgrade back to the current version of gcc-11 (11.4).

Granted, the installation isn't guaranteed to work, could be insecure or lacking security updates. Also, you may run into dependency issues. If that's the case, you'd have to tread carefully or install each dependency one by one but always pay attention to what, if any, packages would need to uninstall when installing weird versions because you don't want to inadvertently uninstall a bunch of stuff you want or need.

EDIT: you will need to install dependencies (included above) but you can run the following commands to do it more quickly.

Run the following commands to create a working directory and download the packages:

cd
mkdir GCC11.3
cd GCC11.3
wget 'http://mirrors.kernel.org/ubuntu/pool/main/g/gcc-11/gcc-11-base_11.3.0-6ubuntu1_amd64.deb'
wget 'http://mirrors.kernel.org/ubuntu/pool/main/g/gcc-11/libgcc-11-dev_11.3.0-6ubuntu1_amd64.deb'
wget 'http://mirrors.kernel.org/ubuntu/pool/universe/g/gcc-11/gcc-11_11.3.0-6ubuntu1_amd64.deb'

Now, install the packages. Again, however, remember to pay attention to what, if any packages will be uninstalled when you install these weird versions. Review the list before you select Y to accept the changes!:

sudo apt install ./gcc*.deb

And again, when you're done compiling or whatever you need gcc 11.3 for, simply run:

sudo apt install --reinstall gcc-11 gcc-11-base libgcc-dev

to upgrade back to gcc 11.4.

Major versions add new features. Minor versions fix bugs. If a bug is found in an earlier major version, a new minor version is released to fix it.