"Sick Of You" / Lindsey Released A Song He Didn't Produce?
 

With the exception of Buckingham Nicks I've never heard of Lindsey releasing music he didn't produce or co-produce himself, but IMDb credits production solely to Jon Brion: https://www.imdb.com/title/tt1758830/soundtrack

Can't say I much care for the song after listening to it on YouTube.

Did he really let someone else produce him, and are there other examples of his doing so?

Jon Brion is an amazing producer.
He did Fiona Apple’s best album, Rufus Wainwright’s best album,
Elliot Smith’s best album, and Aimee Mann’s best album.
I’d love to see more collaboration between these two.

Yes, amazing at so much. I saw him 100 times when I lived in L.A. (he had a weekly residency).

He never produced an Elliott album. He plays on XO and sings on Figure 8, but produced neither. I believe he did, however, produce one Elliott song ("True Love") that posthumously went on a soundtrack to a documentary of Elliott himself.