We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs $\widehat{G}_{μ,ν}$, which we call clique-independent graphs, indexed by two compositions $μ$ and $ν$. Moreover, we define a delay statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the celebrated shuffle theorem of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials $\langle \nabla e_n, e_μh_ν\rangle$ in terms of these configurations.