Often when I launch a game through Steam that “processing Vulkan shaders” window appears and loads for a couple minutes. Sometimes it takes no time, sometimes it takes several minutes. But then, for larger games like Dune Awakening or Outer Worlds 2, the game needs to sit and process shaders for another couple minutes anyway. But for some games, like Enshrouded, I can skip the Vulkan processing with no problems in the game (I do that because the Vulkan processing doesn’t go anywhere). So what is that Vulkan processing for?
It wouldn’t surprise me if it processed vulkan shaders briefly every time, to check for recently used ones and upload them to Steam.
If it’s taking more than a few seconds every time, I would guess that either your last play session triggered a bunch of new shaders (for example if you explored a new area that used different visual effects), or you’ve stumbled upon a bug.
If you think it’s a bug, you should consider checking for reports on Valve’s issue tracker entry for that game, or on the Steam for Linux issue tracker if it affects many games. If there are no reports already, consider creating one.