We study a Majorana island coupled to a bulk superconductor via a Josephson junction and to multiple external normal leads. In the absence of the Josephson coupling, the system displays a topological Kondo state, which had been largely studied recently. However, we find that this state is unstable even to small Josephson coupling, which instead leads at low temperature T to a new fixed point. Most interesting is the case of three external leads, forming a minimal electronic realization of the long sought two-channel Kondo effect. While the T=0 conductance corresponds to simple resonant Andreev reflection, the leading T dependence forms an experimental fingerprint for non-Fermi-liquid properties.