We have constructed a sample of 29 close projected quasar pairs where the background quasar spectrum reveals absorption from optically thick HI gas associated with the foreground quasar. These unique sightlines allow us to study the quasar circumgalactic medium (CGM) in absorption and emission simultaneously, because the background quasar pinpoints large concentrations of gas where Ly-a emission, resulting from quasar-powered fluorescence, resonant Ly-a scattering, and/or cooling radiation, is expected. A sensitive slit-spectroscopic search (1-sigma limits of SB_Lya ~= 3e-18 erg/s/cm^2/arcsec^2) for diffuse Ly-a emission in the environments of the foreground quasars is conducted. We fail to detect large-scale ~ 100 kpc Ly-a emission, either at the location of the optically thick absorbers or in the foreground quasar halos, in all cases except a single system. We interpret these non-detections as evidence that the gas detected in absorption is shadowed from the quasar UV radiation due to obscuration effects, which are frequently invoked in unified models of AGN. Small-scale R_perp <~ 50 kpc extended Ly-a nebulosities are detected in 34% of our sample, which are likely the high-redshift analogs of the extended emission-line regions commonly observed around low-redshift (z < 0.5) quasars. We also detect a compact high rest-frame equivalent width (W_Lya > 50 A) Ly-alpha-emitter with luminosity L_Lya =2.1+-0.32e41 erg/s at small impact parameter R_perp=134 kpc from one foreground quasar, and argue that it is more likely to result from quasar-powered fluorescence, than simply be a star-forming galaxy clustered around the quasar. Our observations imply that much deeper integrations with upcoming integral-field spectrometers such as MUSE and KCWI will be able to routinely detect a diffuse Ly-a glow around bright quasars on scales R ~ 100 kpc and thus directly image the CGM. [abridged]
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