AWS has put work into fast-starting containers [1] using tricks like lazy loading container storage, profiling container startup, non-lazily priming critical blocks, and caching shared blocks. IIRC parts of it are open source. I don't know if enough of it is open source to be helpful, but it's cool stuff!

[1] Gigabytes in milliseconds: Bringing container support to AWS Lambda without adding latency. https://www.youtube.com/watch?v=A-7j0QlGwFk