Have you checked https://deployer.org? And there is a package by Loris Leiva that have good default configuration integration with Laravel: https://github.com/lorisleiva/laravel-deployer

Never heard of deployer. It seems like an interesting project, I will check it out. On first glance seems very similar to Laravel Envoy.

I am also considering to use combination of Forge + Envoyer: creating new server for each customer and then manage deployments with Envoyer. I would like to have a web dashboard to manage all the deployments...

I am not sure that having separate server for each customer is overkill or not, at the end system should be easy to manage even with 1000 customers.