Suitors flew from California and Washington to the Upper East Side in pursuit of Ina Garten, also known as the Barefoot Contessa. They brought gifts, including bags of candies and a jar of homemade pickles. What they were after was a piece of the beloved Food Network star’s forthcoming foray into podcasting, a move that prompted a bidding war and secured her a seven-figure deal with Vox Media — an unusual level of interest for a yet-to-be-made show.