The intrinsic Lyalpha emission lines of normal galaxies before reionization are much absorbed by the dampingl wing of the Gunn-Peterson trough, rendering their direct detection nearly impossible, if their intrinsic line widths are less than similar to100 km s(-1). High-redshift luminous quasars prior to the completion of cosmological reionization at z similar to 6, on the other hand, are capable of producing large H II regions around them (Stromgren spheres) to allow their intrinsic Lya emission lines to be transmitted without overwhelming absorption (Cen & Haiman). We suggest that targeted observations at the Stromgren spheres of known luminous quasars at z greater than or equal to 6 would be able to detect Lya emission lines of galaxies inside the Stromgren spheres largely unattenuated. A tunable, very narrowband filter of (Deltalambda/lambda) similar to 0.1% or a narrowband filter of (Deltalambda/lambda) similar to 1% with follow-up spectroscopic identifications will be required. Such observations could directly observe the sources of cosmological reionization including possibly the Population III galaxies at z = 6-20 by the James Webb Space Telescope. Possible applications include determinations of the ionization state of the intergalactic medium, the sizes of the Stromgren spheres, the ages of the quasars, the luminosity function of high-redshift galaxies and its evolution, the spatial distribution of galaxies and its evolution, the biased distribution of galaxies around quasars, and the anisotropy of quasar emission. Observations using Keck-class telescopes may already be made to enable a differentiation between a fully neutral and a 10% neutral intergalactic medium at z > 6.