The need for higher energy-density rechargeable batteries has generated interest in alkali metal electrodes paired with solid electrolytes. However, metal penetration and electrolyte fracture at low current densities have emerged as fundamental barriers. Here we show that for pure metals in the Li--Na--K system, the critical current densities scale inversely to mechanical deformation resistance. Furthermore, we demonstrate two electrode architectures in which the presence of a liquid phase enables high current densities while it preserves the shape retention and packaging advantages of solid electrodes. First, biphasic Na--K alloys show K+ critical current densities (with the K-$\beta${\textacutedbl}-Al2O3 electrolyte) that exceed 15{\thinspace}mA{\thinspace}cm‒2. Second, introducing a wetting interfacial film of Na--K liquid between Li metal and Li6.75La3Zr1.75Ta0.25O12 solid electrolyte doubles the critical current density and permits cycling at areal capacities that exceed 3.5{\thinspace}mAh{\thinspace}cm‒2. These design approaches hold promise for overcoming electrochemomechanical stability issues that have heretofore limited the performance of solid-state metal batteries.