Well, Mudguard, your site says:

"Already in 1325 the eagle was added as the supporter behind the shield. The eagle was the Imperial Eagle as the Dukes of Austria, from the Habsburg dynasty,were Kings (and Emperors) of the Holy Roman Empire at the time. Ever since the Habsburg family has used the eagle, first one-headed, later two-headed as the main supporter."

So, that tells us that initially, the eagle had only one head. It does not say where the other head came from, or when.

I don't know that much about mediaeval history, but if the Habsburgs were Emperors as early as 1325, they must have been out of office for a substantial amount of time in the 15th century.