We analyze a family of graphs known as banana graphs, with two marked vertices, through the lens of Hurwitz-Brill-Noether theory. As an application, we construct explicit new examples of finite graphs which are Brill-Noether general. These are the first such examples since the analysis of chains of loops by Cools, Draisma, Payne and Robeva. The graphs constructed are chains of loops and "theta graphs," which are banana graphs of genus 2. We also demonstrate that almost all banana graphs of genus at least 3 cannot be used for this purpose, due either to failure of a submodularity condition or to the presence of far too many inversions in certain permutations associated to divisors called transmission permutations.