The URI will be incorrect for the delete action; you are posting to admin.administrator.destroy rather than to admin/administrator/123 (where 123 would be the $this->user->id.
The correct form for the URI is admin/administrator/123, which you can get using the route helper:
route('admin.administrator.destroy', $this->user->id)
Rather than making a test postrequest, you can make a delete request:
$this->delete(route('admin.administrator.destroy', $this->user->id))
->assertViewIs('admin.index');