Okay, so.. just because it took me ages to undestand (mainly because I am new to laravel) I want to pinpoint the problem that took me two days to find...
So basically I am using Homestead + gitHub + Forge on a digitalOcean droplet.
My homestead installation and new project creation went fine, the local server is up and running and laravel is working.
GitHub is linked with forge and deploy automatically.
YET !
I was having this error : whoops, looks like something went wrong.
The problem was : .env not commited to gitHub
Explanation : .env is not commited to gitHub because holds sensitive information. It is initially set in your .gitignore file.
How to fix it : copy and paste your local .env text into your forge site through the web app (environnement panel) - change access codes for the Db.
Ik, all the pros are laughing at me right now :)