In the early 2000s, Frigerio, Martelli, and Petronio studied 3-manifolds of smallest combinatorial complexity that admit hyperbolic structures. As part of this work they defined and studied the class Mg,k of smallest complexity manifolds having k torus cusps and connected totally geodesic boundary a surface of genus g. In this paper, we provide a complete classification of the manifolds in Mk,k and Mk+1,k, which are the cases when the genus g is as small as possible. In addition to classifying manifolds in Mk,k, Mk+1,k, we describe their isometry groups as well as a relationship between these two sets via Dehn filling on small slopes. Finally, we give a description of important commensurability invariants of the manifolds in Mk,k.