Good restaurant with live music, or show for 35th birthday diner?
My favorite for good music and really good food is Uncommon Ground.
Two locations: one on Devon Avenue in Edgewater and one on Clark St. in Lakeview. Affiliated with the Old Town School of Folk Music and a Member of the Slow Food Movement, Uncommon Ground has an organic rooftop garden at the Devon Ave. location.
The websites will give you the music scheduled for the date of your event and you can also check out the menus.
Up north there are two places where a restaurant is in front of a music venue, but they don't do the music where the food is served; they're probably not exactly what you have in mind, but I'll describe them anyway.
Act One Pub is the restaurant in the front of Mayne Stage in Rogers Park, and the food is excellent. But you can't take the food inside to the show.
Union Pizzeria is the restaurant in the front of Space in Evanston; I'm not a fan of their pizza or other food. You're permitted to take food from Union into Space, but the tables in the reserved seating are tiny, so you may be juggling the pizza on your lap. And you'll definitely be doing that if you're in the general admission seats.
Both venues have their music schedules on their websites, so you can pick a show for whatever music pleases you most.