Spitfire Audio have a good interview with Jay Wadley, the composer behind Ending Things' musical score.

How did you land the I’m Thinking of Ending Things gig? 

Anthony Bregman [a producer on I’m Thinking of Ending Things] is the one who reached out to me. He and I had worked together before on the film Indignation back in 2016, so that's how we had gotten to know each other. He was just like “can I interest you in a Charlie Kaufman movie?” And my jaw dropped and I probably passed out. I did not see that coming. There were a lot of very specific things that he wanted in the score that he felt were really uniquely suited to my experience and my background, which was pretty cool.

I read the script and then had a call with Charlie Kaufman, just to start to get some ideas together as to what sort of approach we’d want to take with it and it seemed like we were on the same page pretty quickly. We had to move fast because I had to write a bunch of stuff prior to shooting. So the ballet and the sort of 1950’s style, weird jingle that they sing on camera – all of that stuff had to be completely written prior in order to shoot with it.

[...] This score demanded a very particular approach, and I wouldn't want to try to live in the world of Jon Brion or Carter Burwell and try to do something that would sound like them. I wanted it to be wholly my own unique voice to bring my own sensibilities to it because I think if I had tried to, you know, do what they do for a Kaufman film, it would be very different. (Source)