  The intense radiation emitted by luminous quasars dramatically alters the
ionization state of their surrounding IGM. This so-called proximity effect
extends out to tens of Mpc, and manifests as large coherent regions of enhanced
Lyman-$\alpha$ (Ly$\alpha$) forest transmission in absorption spectra of
background sightlines. Here we present a novel method based on Ly$\alpha$
forest tomography, which is capable of mapping these quasar `light echoes' in
three dimensions. Using a dense grid (10-100) of faint
($m_r\approx24.7\,\mathrm{mag}$) background galaxies as absorption probes, one
can measure the ionization state of the IGM in the vicinity of a foreground
quasar, yielding detailed information about the quasar's radiative history and
emission geometry. An end-to-end analysis - combining cosmological
hydrodynamical simulations post-processed with a quasar emission model,
realistic estimates of galaxy number densities, and instrument + telescope
throughput - is conducted to explore the feasibility of detecting quasar light
echoes. We present a new fully Bayesian statistical method that allows one to
reconstruct quasar light echoes from thousands of individual low S/N
transmission measurements. Armed with this machinery, we undertake an
exhaustive parameter study and show that light echoes can be convincingly
detected for luminous ($M_{1450} < -27.5\,\mathrm{mag}$ corresponding to
$m_{1450} < 18.4\,\mathrm{mag}$ at $z\simeq 3.6$) quasars at redshifts
$3<z_\mathrm{QSO}<5$, and that a relative precision better than $20\,\%$ on the
quasar age can be achieved for individual objects, for the expected range of
ages between 1 Myr and 100 Myr. The observational requirements are relatively
modest - moderate resolution ($R\gtrsim750$) multi object spectroscopy at low
$\rm{}S/N > 5$ is sufficient, requiring three hour integrations using existing
instruments on 8m class telescopes.
