Replace profile link with avatar dropdown in header
Add an avatar-triggered dropdown menu containing "My profile" and "Log
out" so authenticated users can sign out from any page — previously the
sign-out button only existed on the profile page itself, which was
inaccessible during first-lookup backfill. Avatar is resolved via
AtprotoClient and cached in memory for 1h per user.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>