We study the allocation of m indivisible chores among n agents with additive disutilities under the fairness notion of envy-freeness up to one chore (EF1) and the efficiency notion of Pareto-optimality (PO). Although the existence of an allocation satisfying both EF1 and PO was recently established using a highly non-constructive fixed-point argument, an effective algorithm for computing such an allocation remains elusive, prompting the study of meaningful relaxations of these desiderata. Prior work introduced a natural relaxation through the concept of epistemic fairness: an allocation is said to be epistemic EF1 (EEF1), if for every agent i, it is possible to re-allocate the bundles of agents other than i such that i becomes EF1. In this work, we present a pseudo-polynomial time algorithm for computing an allocation of chores that is both EEF1 and PO. This gives an efficient polynomial-time algorithm for most practical settings where disutility values are integral and polynomially bounded in m and n. Our result employs the competitive equilibrium framework and relies on several technical insights that both utilize the distinct structure of epistemic EF1 and address the challenges it introduces.