The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be $σ$-avoiding, for some permutation $σ$, meaning that, at each step, the elements maintained in the stack avoid the pattern $σ$ when read from top to bottom. Since the set of permutations which can be sorted by such a device (which we call $σ$-machine) is not always a class, it would be interesting to understand when it happens. We will prove that the set of $σ$-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we will analyze two specific $σ$-machines in full details (namely when $σ=321$ and $σ=123$), providing for each of them a complete characterization and enumeration of sortable permutations.