If you use RahnoMock framework for Mocking then it is possible. You can visit the Link
But my personal opinion is Abstract should be always abstract. In your custom base controller is an abstract class. So it should not be testing directly. Why because it is an abstract so no object creation is possible. What you can do you can create a subclass of that base controller for only unit test(subclass might be exists to your unit test project) and can follow do nothing trick and use this all your test places. But again though Mocking Framework provides the feature so others opinion might be there. I say only my personal opinion.