How would you approach making immutable live-cd like Linux? No persistence at all, just boot it and run some app - think some kind of presentation panel which shows predefined program/URL. Ideally net booted to avoid having storage at all.
Ideally, Kali is meant to be used this way, so you can use it to perform forensics on a potentially compromised system without inadvertently changing anything and destroying evidence. You can really trivially do this with any Live CD by putting it on write-once read many media. You typically need at least /var and /tmp to be writeable, but that can be accomplished via tmpfs so they only write to memory and not disk. You don't really need to do anything to enable this at the distro level, other than maybe make the mounts default, but in practice, live CDs tend to mount the root filesystem as SquashFS, which is also read-only at the filesystem level, and then use OverlayFS for partitions that need to be writeable for software to work, not retaining the writeable layer on shutdown.
If you mean how would you do it yourself, you can use the tooling used by real distros. I'm not sure what tools they all provide, but Archiso (https://wiki.archlinux.org/title/Archiso) is probably the simplest to understand and modify because it's purely shell scripts.