It was always interesting going to Red Robin with a Hindu vegetarian. Not only was meat not on the table, but with a strenuous objection to beef. No objection to anyone else, but man that was odd. His only option was the veggie burger at the time, although I'm not sure exactly what it was. Might have been mostly black beans. I'm not even sure they reserved a part of the grill just for that. Even the Hindus I knew of that had no objection to chicken might have been upset if the grill was used for beef, or others like Muslims who might object if the same grill was used for bacon. I've worked with lots of people from different backgrounds, and finding places that would be fine for everyone could be tricky.
Burger King doesn't cook the Impossible Whopper on a separate setup except on request. They'll clean off an area to do that. Not sure about Red Robin. They claim they have vegetarian/vegan options, which sounds like a separate area, but I guess one should ask to make sure.