What is the name of the inn where Lorelai and Sookie first work together?

The inn where Lorelai and Sookie first work together is the Independence Inn.

Lorelai manages the Independence Inn in Stars Hollow before opening the Dragonfly Inn with Sookie St. James. Sookie serves as the inn's chef, making the two women professional colleagues as well as close friends.

The Independence Inn is an important early setting because it shows Lorelai's hospitality career and establishes Sookie's culinary talent. It also provides the background for several events involving Rory, including the inn's role in Lorelai's life before she becomes an owner.

The Dragonfly Inn is a later business venture and should not be used as the answer to this question. Lorelai and Sookie eventually leave the Independence Inn to create their own establishment.