Using the theory of hyperbolic manifolds with totally geodesic boundary, we provide for every $n\geqslant 2$ a class $\calM_n$ of such manifolds all having Matveev complexity equal to $n$ and Heegaard genus equal to $n+1$. All the elements of $\calM_n$ have a single boundary component of genus $n$, and $\#\calM_n$ grows at least exponentially with $n$.