Thanks to a brew package by FiloSuttile, and a post by Timryan, it’s actually pretty easy.
# Install the necessary C cross-compiler
# This takes a while
brew install FiloSottile/musl-cross/musl-cross
# Install the rust target you want
rustup target add x86_64-unknown-linux-musl
Now you need to change your Cargo configuration to use your newly installed musl linker
# ~/.cargo/config
[target.x86_64-unknown-linux-musl]
linker = "x86_64-linux-musl-gcc"
Now you should be able to compile your binary
# Build your app for that target
cargo build --release --target x86_64-unknown-linux-musl
# Your build is in /target/x86_64-unknown-linux-musl/release/your-gizmo