We present an overview of the GBT Observations of TMC-1: Hunting Aromatic Molecules Large Program on the Green Bank Telescope. This and a related program were launched to explore the depth and breadth of aromatic chemistry in the interstellar medium at the earliest stages of star formation, following our earlier detection of benzonitrile (c-C6H5CN) in TMC-1. In this work, details of the observations, use of archival data, and data reduction strategies are provided. Using these observations, the interstellar detection of propargyl cyanide (HCCCH2CN) is described, as well as the accompanying laboratory spectroscopy. We discuss these results, and the survey project as a whole, in the context of investigating a previously unexplored reservoir of complex, gas-phase molecules in pre-stellar sources. A series of companion papers describe other new astronomical detections and analyses.