To unlock the Shroomtender Gloves, you need to complete the Necrom Cave Delver achievement. There's a good chance you are familiar with versions of this achievement from other areas in ESO.
To finish it, you need to find and clear every Delve in the Telvanni Peninsula and Apocrypha zones. Completing a delve normally means making your way to the main boss and defeating them.
This achievement is not likely to cause you any real issues and most players should be able to solo it quite easily as long as you have a few levels up and a full set of gear.
Below are the locations of the six delves that you need to clear. Four of the delves are in Apocrypha, and two are on the Telvanni Peninsula.
ESO Apocrypha Delve Locations
ESO Telvanni Peninsula Delve Locations
