Concurrent change of employer and residency status

Hello;

I currently hold a 5-year EEA residence permit (my partner of 15 years is French). We have both lived in the UK for ten years. My current permit expires in a few weeks, and I was going to apply for a permanent residence card. However, I have just been offered a job with a new employer, which is to start January 7 (i.e. less than two months). My current employer was fine to continue employing me as long as my application was in process, but new new employer needs to see a valid the residence permit in my passport.

Is there any way to fast-track an application? Is it *actually* necessary for the new employer to have a valid permit, or is proof that the application is underway enough?

Many thanks for any advice.

Seanna