Yes you can and start whatever program you want, take i look at better-initramfs (https://github.com/slashbeast/better-initramfs), although it is more for gentoo/funtoo.

Basicaly when linux boot from initrd it starts "/init" executable and chroots to your system (ok, it is using pivot_root and it is slightly more complex), but idea it is same as manually mounting filesystems and doing chroot then starting executable /bin/init