Terry Rockafellar has a paper on this topic in SIAM Optimization titled [*Generalized Hessian Properties of Regularized Nonsmooth Functions*][1]. See theorem 3.8. Rockafellar proves an equivalence between twice-differentiability of the Moreau envelope and twice epi-differentiability of the function $f$.


  [1]: https://doi.org/10.1137/S1052623494279316