Keeping in mind that Biryani comes in a staggering variety of regional, cultural, and national variations, I'll offer that Hyderabad House does a pretty spot on Hyderabadi Biryani - intensely spicy from red chili, sour from tamarind, and fairly light on the stronger aromatic qualities of some other Biryanis (e.g. Awadh/Lucknow Biryani and other Biryanis of India's North which often feature strong rosewater, saffron, cardamom, mint and other flavors).
Hyderabadi Biryani basically adapts the Mughal Biryani (itself a dish with Persian roots) to the local Telegu tastes; again, spice, sour, salt. Hyderabad House's Biryani, to me, is a pretty accurate simulacrum of the city's famous Biryani chain Paradise Biryani. Personally, I found Paradise's Biryani to be fairly middling compared to other Biryanis in Hyderabad (see Shah Ghouse or Hotel Nayab), but still damn good. And in Chicago, great.
Making HH's version especially authentic and delicious are the inclusion of the traditional Hyderabadi accompaniments: most often mirch ki salan (peppers in tamarind/peanut masala) or just salan (masala minus the peppers) and of course raita (yogurt sauce). I'd opt for the goat/mutton version. Chicken Biryani is for suckers and fake vegetarians who are afraid of red meat.

I also love Ghareeb Nawaz's Biryani, also prepared more or less in the typical Hyderabadi style with the same accoutrements.
I'd love to hear if anyone has any recs for Kolkata Biryani (an exemplary Northern Biryani with tons of aromatics, still spicy, and spiked with egg and potato), Lucknow Biryani (the king of Biryanis in India according to me) and Kerala Biryani (again, showing how Biryani adapts to its local cuisine - in this case loaded with local favorites coconut milk, powerful cinnamon and clove, and often made with seafood).
Bismillah on Ridge used to make a great generic Mughal style Biryani, but I think they are closed.
For non-South Indian variations, try Zorbian at Yemeni or Somali places, Kapsa or Maqlouba at Middle Eastern places, or Pulao at Afghan or Persian places.
"By the fig, the olive..." Surat Al-Teen, Mecca 95:1"