We discuss expansions of monoids in the class of two-sided restriction monoids. For a monoid $M$ and a class of partial actions of $M$ determined by a set, $R$, of identities, we define $FR_R(M)$ to be the universal $M$-generated two-sided restriction monoid with respect to partial actions of $M$ determined by $R$. This is an $F$-restriction monoid which (for a certain $R$) generalizes the Birget-Rhodes prefix expansion of a group.  We  explain that  $FR_R(M)$ can be coordinatized as a semidirect product of the idempotent semilattice of a similarly defined inverse monoid, partially acted upon by $M$. As a special case, this yields the result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid. We also present some interesting particular cases and examples.