A new scheme for a laser-driven proton accelerator based on a sharply tailored near-critical-density plasma target is proposed. The designed plasma profile allows for the laser channeling of the dense plasma, which triggers a two-stage acceleration of protons—first accelerated by the laser acting as a snowplow in plasma, and then by the collisionless shock launched from the sharp density downramp. Thanks to laser channeling in the near-critical plasma, the formed shock is radially small and collimated. This allows it to generate a significant space-charge field, which acts as a monochromator, defocusing the lower energy protons while the highest ones remain collimated. Our theoretical and numerical analysis demonstrates production of high-energy proton beams with few tens of percent energy spread, few degrees divergence angle and charge up to few nC. With a PW-class ultrashort laser this scheme predicts the generation of such high quality proton beams with energies up to several hundreds of MeV.