11/2021 update (post-launch of Campaign mode and full release): I can confirm the comment by /u/NoteLost is correct; the game works with Proton experimental and no launch options!

Microsoft killed my one Windows PC by trying and failing to update, so I had to resort to finding a way to play on Linux. Evidently, playing this game on Linux is nothing new, but I wanted to write out a guide for what worked for me, to hopefully save you some Googling.

The main part of the guide (steps 2-4) you can just follow here, complete with helpful screenshots: https://itsfoss.com/steam-play/

1. Download Steam to your Linux computer

2. In the Steam menu dropdown, go to Settings and check "Enable Steam Play for all other titles" (here you can also select the Proton version)

3. Install the Gloomhaven game to your library. Click Play if you're feeling lucky, but at this point it errored out for me before the game even started

4. On the Gloomhaven game page, click the gear/cogwheel (hey, like the Tinkerer symbol!) on the right below the banner, then click Properties...

5. Click SET LAUNCH OPTIONS... then paste in PROTON_USE_WINED3D11=1 %command% and click OK. I found this code in a comment here which directly references the Proton documentation -- what it does is set it to use Directx 11, which fixed the error I got when I tried clicking Play the first time

6. If it still doesn't work, try different Proton versions (see step 2). 5.0-9 and 4.11-13 crashed right away without any error messages. Finally at 4.2-9 I got an error message that led me to the fix in step 5. So the latest Proton version could possibly work with that step 5 fix, but I haven't tried.

Edit: Fix step numbers, also co-credit on step 5 to u/KFded on this post -- I had a lot of tabs open :)