Fat Tonys Delicatessen began serving Vienna Beef Hot Dogs (Chicago Style) in Lubbock, Texas in 2010. We have a full menu and serve customers who want to be treated as family and not as animals in a zoo. Big Spring is a growing market due to the recent oil boom.. The brand is important for entrepreneurs seeking to improve their lives through hard work and a love for strangers. The goal is to make stores available at a reasonable price.