Wow cant believe no one has fished Big Bar Lake. Either it sucks or it is great and nobody wants to give up a type of fly to use. Guess I have to go with my old favourite, Dr. Spratley which I have in every color and size.
No I think it's just too general of a question. Big bar is like any other lake in June, unpredictable. Many bugs can and will be available to trout in June, so one should be prepared for anything, not just one favorite or known good fly.
What kind of fly fishing do you plan on doing?
Will you be trolling a fly? And if so what types of lines do you think you will be using?
Will you be anchoring and fly fishing? If so what types of lines do you have available to use?
What date in June? June 1st and June 28th will probably be much different.
Just those 2 questions will change what favorite or known good fly is good in June, and you would probably get 10 different answers based on your answer to the questions.
Chironimids, dragons, damsels, mayflies are good place to start for Big Bar or any other lake in June.