Ha ha ha the best question so far! You don't have to go far in Bihar to find a litti chokha shop. However in Patna, its not that popular contrary to popular belief that all Biharis eat litti chokha. Nonetheless, you'll still find numerous shops and stalls selling it.
The best littis are made on fuel made of cow dung which is more or less made in the villages. Here's a story about Aamir Khan eating the litti chokha when he was in Patna.
A good place where you can find the little chokha in Patna include litti-chokha stall at platform 1 of Patna Junction. In fact, you'll find them everywhere around the station. Kindly watch this interesting video about a street stall selling litti chokha.
Best wishes.