Giant radio lobes forming at both ends of powerful extragalactic jets are often suspected to generate and host some of the highest-energy cosmic rays in the Universe. Using particle-in-cell simulations of a magnetized relativistic jet termination shock, we show that particles are efficiently accelerated up to the confinement limit of the system provided that the global transverse structure of the jet is taken into account. A strong tangential velocity discontinuity grows between the core and the edges of the jet leading to macroscopic shear-flow particle acceleration. Energetic particles accumulate in an over-pressured and under-dense bubble near the shock front before escaping in the downstream medium via a Von Karman vortex street. A possible observational signature of this mechanism would be the detection of an underluminous synchrotron cavity upstream of AGN jet hotspots. These results suggest that extragalactic jet termination shocks provide exquisite conditions to accelerate cosmic rays up to the highest energies.