We define for each $g\geqslant 2$ and $k\geqslant 0$ a set $\calM_{g,k}$ of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus $g$. Manifolds in $\calM_{g,k}$ have Matveev complexity $g+k$ and Heegaard genus $g+1$, and their homology, volume, and Turaev-Viro invariants depend only on $g$ and $k$. In addition, they do not contain closed essential surfaces. The cardinality of $\calM_{g,k}$ for a fixed $k$ has growth type $g^g$.\\ \hspace*{4mm} We completely describe the non-hyperbolic Dehn fillings of each $M$ in $\calM_{g,k}$, showing that, on any cusp of any hyperbolic manifold obtained by partially filling $M$, there are precisely $6$ non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with $\partial$-reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If $M$ has one cusp only, the three $\partial$-reducible fillings are handlebodies.