A while ago, I wrote the userscript You've Earned This Badge, but when and where? which does exactly this:
This userscript turns the checkmarks in the badge list (which indicate you've earned a
specific badge) to links to your personal badge page, where you can see when and where
you earned the badge.
You can install the userscript with
this direct link.
I was finally awarded the Enlightened badge at 14:55 on 4/16, 28.3 hour after becoming eligible. From animuson♦ in comments:
Based on the query, your answer is eligible to be awarded the badge whenever it runs next. The "within 24 hours" thing is true in most cases, but not technically correct anymore. All the badge queries used to run on set ...