How to Install 22.10.X Mesa Drivers?

This currently requires the 22.10 Linux driver from the AMD Radeon repository.

https://wiki.blender.org/wiki/Reference/Release_Notes/3.2/Cycles

Blender officially supports only the proprietary drivers from both AMD and NVIDIA

Blender - ArchWiki

It’s not referring to Mesa, it’s referring to Radeon Software for Linux. See AMDGPU PRO - ArchWiki