Burke vs Chicago Cut?
Both are fantastic and my top 2 in Chicago. If there's interest in the 75 day aged ribeye (or trying that and a lesser aged steak to taste the differences), then DB Primehouse is the call.
I prefer the sides, rooms/ ambiance, and the service at Chicago Cut. And the steaks are fantastic as well. So if looking for a more "rounded" experience, would go there. Have a great time!