We present a new method to directly map the neutral-hydrogen distribution during the reionization epoch and constrain the emission properties of the highest redshift quasars (QSOs). As a tracer of H I, we propose to use the Lyα radiation produced by quasar ionization fronts (I-fronts) that expand in the partially ionized intergalactic medium (IGM) before reionization is complete. These Lyα photons are mainly generated by H I collisional excitations. Combining two radiative transfer models (one for the QSO ionizing radiation and one for the Lyα photons), we estimate the expected Lyα spectral shape and surface brightness (SB_Lyα) for a large number of configurations where we varied both the properties of the ionizing QSO and of the surrounding medium. We find that the expected signal is observable as a single (broad) emission line with a characteristic width of 100-200 km s⁻¹. The expected SB_Lyα produced at redshift z 6.5 within a fully neutral region (at mean density) by a typical QSO I-front lies in the range 10⁻²¹ to 10⁻²⁰ ergs s⁻¹ cm⁻² arcsec⁻² and decreases proportionally to (1 + z)² for a given QSO age. QSOs with harder spectra may produce a significantly brighter emission at early phases. The signal may cover up to a few hundred square arcminutes on the sky and should be already detectable with current facilities by means of moderate- to high-resolution spectroscopy. The detection of this Lyα emission can shed new light on the reionization history, the age and the emission properties of the highest redshift QSOs.