0) Make sure you own the game you want to track the time with on Steam.
1) Install Source Port(s) of choice.
2) Install ZDL https://zdoom.org/wiki/ZDL
3) Add ZDL as a non-steam game.
4) In ZDL's installation folder (the folder with ZDL.exe), create a text file and name it steam_appid.txt. In the body of the text file just type 2280 and save.
5) When you launch ZDL from Steam, Steam will now track Doom + Doom II as long as the ZDL launcher window is open (so make sure you don't close it when you enter the game, the ZDL program is all that is being tracked.).

*For tracking older Doom games, replace 2280 with the steam app id code for those instead:
Doom II = 2300
Master Levels for Doom II = 9160
Final Doom = 2290

**Yes, this works with other games as well. You could set it up so that notepad.exe tracks as Baldur's Gate 3 if you want.