Don't know for sure but would it be that's allright mama I seem to think he did that on a radio show.Should know because I am a big fan [ shame on me] I think there was also Blue moon of kentuckey on the same show.g.
I remember seeing a documentary a few years ago about Elvis.
It did say he was rather shy and wanted to go into a studio and record a song for his mother. The studio offered to record a song for you for a few dollars.
This of course was never released.
This wiki article talks about his first recordings.
Scroll down to "First recordings and performances"
The Elvis album "The Great Performances" has the original recordings of "My Happiness" and "Thats All Right" on it, as well as many other 'greats'. It doesn't give dates however they are track listed 1 then 2 as above, so I don't know if that is a clue to which was first.......rock on, Elvis lives!