Matt Stauffer has a post to his site today sharing a step-by-step guide to setting up your Laravel application for zero downtime deploys when using the Envoyer/Capistrano combination.
The reason you're getting zero-downtime deploy from these tools is because the entire deploy process—clone, composer install, etc.—doesn't happen in the directory that is currently serving your site. Instead, each new release gets its own separate "release" directory, all while your site is still being served from its current "release" directory.
All of these release directories are just subdirectories of releases. Each directory here represents one of your deploys, and each directory individually has everything needed to serve your site. [...] So, once the build process is complete for each new release, your deploy tool will delete the current symlink and create a new current symlink that points to your latest release. Boom. Now that release is live.
He then relates this back to the deployment of a Laravel application with Envoyer which already follows this "symlink deploy" method. There's a few caveats he mentions with this deploy, however, including information that shouldn't be removed in each deploy (like caches or configuration files). He then provides the steps (commands) you can follow with the deploy to manually use the "symlink deploy" method in your own scripting.