Its quite natural for someone to crave for food of a different region in the region where they live. Tamilians ask for Sambar Idly in Delhi. Punjabi's ask for Alu Parantha in Chennai... Most of the time they would not be happy with the food they get but stay contended that they ate what they wanted.
Chennai however is a place which has food for ppl of all states(atleast as far as i know). It also has one of the best Vegetarian Punjabi Restaurants. The Desi Rasaoi's Gyans Vaishnav Dhaba has the best veggy punjabi food. There are wide variety of Salads, Starters, Paranthas, Curries and Rice items.. In addition to these, you have the limited thalis (Jain, Punjabi etc). The best part of the meal are the desserts. The kheer and the fruit salad are simply out of the world.
Try out the panneer tikka, alu parantha, chana pulao and boondi raita... The prices might be a tad expensive, but the experience is totally worth it.