We had a great experience at Black Tap in Nashville. They have gluten-free Udi's buns, so any sandwich can altered. Since they're mostly a burger, sandwich, beer and milkshake place, this is about as much as you can ask of the place. Everything was high quality and the service was fast and friendly and helpful.
They do not have a dedicated fryer, so the server (Moses) was careful to make sure we understand that fries and chips would be a risk of cross-contamination. He also let us know about sauces that might be a risk. For our family, we haven't (yet) had issues with fryers, so we decided to eat both the chips and fries. Neither my daughter nor I, both Celiac's, had any issues with eating the fries. Obviously others will want to make their own choices.
I had a burger on an Udi's bun, it was delicious. My daughter does not like burgers, she opted for a chicken sandwich, also on an Udi's bun. She was able to order one of the sandwiches usually with fried chicken to be made with grilled chicken instead. It was also delicious, and she had no issues.
Our non-celiac's family members loved loved their food, as well.
We did not try any of the drinks or desserts, but it did seem like there are standard milkshakes, in addition to their fancy ones, that are likely gluten-friendly at least. We will be going back to try dessert sometime.
Highly recommend.