Do you have the Achievement for completing Varen's Watch (i.e. beating the boss)? This should be what make it turn white (completed), for any character WHO IS NOT USING ADDONS THAT CHANGE THIS.

It sounds like you're actually using the addon named "Character Zone Tracker" which cosmetically reverses the "account wide" nature of Delves, World Bosses and World Incursions (e.g. dolmens) by tracking them on a per-character basis, and overriding the *appearance* on the map by making them look "white" (completed) or "black" (found but non-completed) for any character that hasn't completed them, even when you have the account-wide Achievement for completing them which would under normal circumstances make it look "white" (completed) for everybody.

If you are, indeed, using this addon: then a small number of Delves are currently bugged in it, and its maintainer has retired and nobody has been interested in picking it up. Varen's Watch is one of the delves that is currently bugged in this way: the addon doesn't recognise its completion, even on your *first* character (the one that actually gained the Achievement for completing it), so it will perpetually appear "black" ("found but non-completed") on your map. This may, or may not, be something to do with the fact that it has a dual boss consisting of two monsters rather than one: multi-boss delves are a thing that causes known problems with this addon.

The good news is, if you are indeed using "Character Zone Tracker" addon, you can manually set any such Delve, World Boss or World Incursion to "complete" or "incomplete" by clicking on the icon on your map, and selecting the option to "Beat the Boss at (Delve)" or "Reset the Boss at (Delve)".

It won't, in any way, affect your in-game Achievements for having completed it or not completed it (and if you are using this addon, and turn it off, then the display should reset to correct based on your Achievements): it's a cosmetic thing that allows you to fix the bug of whether you have or haven't completed a given thing with a given character, if the addon fails to set it to completed when you have completed it (or, conversely, sets it to completed when it isn't.)

(Several of the other bugs in this addon are in Craglorn and Cyrodiil's multi-boss delves, and the Anka Ra World Bosses there too. I have already reported them in the addon's thread on ESOUI. Other problems have been noted with the two delves that have multiple entrances - Kings Haven Pass (Summerset) and Gloomreach (The Reach / Blackreach Arkthzand), and a couple of other delves that have boss-level enemies that are not the official delve boss. In High Isle and Galen, I have occasionally noticed that in the interim period between Volcanic Vents after one has just been completed, another one will flash a "false alarm" on the map for a second before a new real vent opens somewhere else, and sometimes this "false alarm" will cause CZT to set a Volcanic Vent to completed if you're at it, even if it never actually blows up. The addon was a somewhat buggy work in process when its owner retired.)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If you are NOT using this addon, then I can't help you.

I hate to ask the question, but you did kill both bosses, right?

Sometimes that happens to me in some zones. I do the exact same quest or objective on another toon and it works fine. I "think" it has something to do with another quest or something taken or completed beforehand that interferes with it.