I'm currently reading Chaos: Charles Manson, the CIA, and the Secret History of the Sixties (it's really good, so far) and at one point the author is interviewing Terry Melcher who boasts about being the "only American to produce the Beatles!". I couldn't find anything by the Beatles that he had produced, although he seems to have been working with Apple in some capacity for a while:

Toward the end of the '60s, Terry Melcher was hired as an independent producer by the Beatles' Apple label. While working for that company, he produced several singles for a fantastic but largely forgotten group called Grapefruit whose primary songwriter was the older brother of Malcolm and Angus Young of AC/DC.

The first Grapefruit single "Dear Delilah" was produced by Terry Melcher when he came to the UK to sign a publishing deal with Apple.

"Lullaby" was produced by John Lennon and Paul McCartney but didn't end up as Grapefruit's second single because Lennon and McCartney was in India at the time. The version found on the album Around Grapefruit was a later, group-produced version that had been remixed by Melcher, who also added an orchestral arrangement. The Lennon/McCartney production wasn't released until 2016.