Okay, the Mycroft-as-emotional-laboratory tag on AO3 is your absolute best bet, and I'm not just saying that. You want character growth that feels earned, not just a tweed-clad robot learning to hug. Filter for the 'Mycroft Holmes & Sherlock Holmes Relationship' tag first—the platonic & sibling dynamic is where the real magic happens for his growth, honestly. Then add 'Angst with a Happy Ending' and 'Character Study' to your filters. Avoid 'Case Fic' unless you want the emotional arc buried under a murder plot.
A specific writer, HiddenLibrarian, does this thing where they write Mycroft's internal monologue as a series of classified file headings that slowly break down. It's a masterclass in showing growth through a changing bureaucratic lexicon. Their fic 'The Unclassified Affair of the Heart' ruined me for a week. Also, don't overlook the 'Sherlock (TV)' fandom's old kinkmeme fills on Dreamwidth; some of the most raw, early-series explorations of Mycroft's loneliness are archived there, full of imperfect but deeply felt character work.
Honestly, half the fun is the hunt. Sort by bookmarks, and you'll find the gems where Mycroft's growth is tied to protecting Sherlock in a way that costs him personally, which is the ultimate catalyst for change in his character.