A scheme for generating high-flux angularly uniform proton beams with high laser-to-proton energy conversion efficiency is proposed. Three laser beams are focused on a microwire array attached to a solid-density hemispheric target. The laser-driven hot electrons from the front of the microwire hemisphere generate a hot-electron sheath in the hollow behind it, so that the protons on its back are accelerated by target normal sheath acceleration. The accelerated protons are of high flux, as well as angularly and energetically uniform. The scheme should be useful for applications involving warm dense matter, such as isochoric heating and modification of materials, as well as for proton therapy and inertial confinement fusion.