To explain the situation a bit:
I'm building an iOS application that uses SSL pinning. I've created a self-signed certificate authority that issues SSL certificates to my web server, and the CA's certificate is bundled with the application for verification. I'd like to use letsencrypt to create the SSL certificates for the web server so that they are trusted implicitly by web browsers, but their certificates wouldn't be signed by my CA so this wouldn't work in the application. (It's worth noting that certificates issued by letsencrypt are very short lived, so they can't be used directly for SSL pinning).
So I'd like to generate a certificate using letsencrypt and then cross-sign it with my CA. Is this possible?