From what I understand from the question you want to get the role/permission that corresponds to your current user.
You will not be able to use laravel (blade) components in react or vue files.
It would be a good idea that when you share the user to your AuthenticatedLayout.jsx file, share it something like: $user = Auth::user()->with('roles', 'roles.permissions'), in this way you will have the user along with relationships to its role and nested permissions to the role. Through the relationships you declare in your User model.
There is also https://inertiajs.com/ that makes a lot of these things easier. I recommend the laravel package permission: https://spatie.be/docs/laravel-permission/v5/introduction.
I hope I helped you, I'm still pending!