The rapid assembly of the massive black holes that power the luminous quasars observed at z ~ 6-7 remains a puzzle. Various direct collapse models have been proposed to head-start black hole growth from initial seeds with masses~105M⊙, which can then reach a billion solar mass while accreting at the Eddington limit. Here, we propose an alternative scenario based on radiatively inefficient supercritical accretion of stellar-mass holes embedded in the gaseous circumnuclear discs (CNDs) expected to exist in the cores of high-redshift galaxies. Our sub-pc resolution hydrodynamical simulations show that stellar-mass holes orbiting within the central 100 pc of the CND bind to very high density gas clumps that arise from the fragmentation of the surrounding gas. Owing to the large reservoir of dense cold gas available, a stellar-mass black hole allowed to grow at super-Eddington rates according to the 'slim-disc' solution can increase its mass by three orders of magnitudes within a few million years. These findings are supported by simulations run with two different hydro codes, RAMSES based on the Adaptive Mesh Refinement technique and GIZMO based on a new Lagrangian Godunov-type method, and with similar, but not identical, sub-grid recipes for star formation, supernova feedback, black hole accretion and feedback. The low radiative efficiency of supercritical accretion flows are instrumental to the rapid mass growth of our black holes, as they imply modest radiative heating of the surrounding nuclear environment.