You could guess by looking back historically: what happened on 6th June 1006, or better still 606AD (06:06:06 06-06-0606)?

You may have to adjust for the change from Julian to Gregorian, but I think that's only a few days, and stuff wasn't often recorded to the day back then.

Random netsearch quotes:

"Chien-chih Seng-ts'an Third Zen Patriarch (606AD)"
"..the organizational entity that calls itself the Catholic Church started in 606AD.."
"..more Patriarchs were added in the East, but after the same order and design of Constantinople in 606AD.."
"Jianzhi Sengcan, the Third Ancestor [died 606AD]"
"Birth of astronomer Aryabhatta 606 AD"
"606 AD - Aedh, son of Colgan, chief of Oirghialla and of all the Airtheara, died on his pilgrimage, at Cluain Mic Nois (Clonmacnoise)"
"The first known independent Bengal king was Shashanka (circa 606 AD)."
"the (Chinese) written exam system of civil service became established in 606 ad"
"the death of Prabahakar Vardhan in 606 AD"
"The City (Cork) was founded on a monastic settlement associated with St. Finbarr in 606 AD"
"The Flavigny abbey , erected in the 7th century ( some say 606 AD ) took every 11th basket at harvest time..."

Can't guarantee that any of these happened on June 6th, at just after 6 in the morning, but it looks pretty likely, doesn't it?