The restaurants in the Loop are accustomed to pre-theater diners and their need to complete dinner quickly. An hour is not a problem at all, and you can comfortably eat even a conventional three-course meal without feeling rushed; just let them know upon your arrival what time you need to be done. Don't be surprised if many of your fellow diners leave en masse at about the same time; it's actually rather humorous to see!
A couple of weeks ago I had an outstanding dinner at tesori, a contemporary Italian restaurant about three blocks from the Bank of America Theater where the Book of Mormon is playing. They're particularly accustomed to meeting tight time constraints because they are in the same building as the Symphony. You can read the detailed report on my terrific meal there, which I posted at www.chow.com/topics/873519#7836347
More excellent restaurants within three blocks of the B of A Theater include Vivere (contemporary Italian, one of the three restaurants in the Italian Village mentioned above), the Florentine (contemporary Italian), the Berghoff (German/continental, mentioned above, no need to relegate yourself to the bar unless that's where you prefer to eat), and Atwood Cafe (contemporary American).
All of these are reasonably priced with entrees no more than the high twenties and, at the Italian places, pasta dishes in the teens (the Florentine is a bit more than the others). Since this is Restaurant Week, you'll also have the option of the $33 or $44 Restaurant Week menu at those restaurants which are participating (all of the above except Atwood Cafe). All of these show availability on Opentable.com for 6:00 seatings this evening. And as noted above, all are within a five-minute walk of the theater.
Enjoy your meal, and enjoy the show!
Thanks for the rec, we went to Tesori. I told them about the show, and they were very good about quick service and getting us in and out on time (he even comped the dessert because it "was taking too long").
I have to say the food was good, though not excellent (though this could just be because the last Italian restaurant we went to was Piccolo Sogno - which was excellent). The veal meatballs were delicious - as was the cheesy polenta that it came with. The beet salad was solid, but could have used some citrus or some acidity to give it more depth. The goat cheese that came with it was very good though. The tortellini was very good, but the paparadelle was just solid. We weren't that thrilled with the almond cake dessert, but we are not really dessert people (though we tend to order it).
Overall, this is a good place, with excellent service, and is great for pre-show dinners.