As an example: suppose you have a personal project like Angular Seed (which is a starting point for an Angular project https://github.com/angular/angular-seed).
Now
What you could do is:
That way, your new project has:
My issue with that is that a seed project isn't a library. It doesn't live in a subdirectory of a project. It IS the project when you start, and you build from there.
Then a simple clone that you push back to your new repo is enough.
But that won't keep any "fork" relationship between your two GitHub repos.
You will have to pull one and push to the other through a local clone.