The Synchrony of Production & Escape: Half the Bright Ly$\alpha$
  Emitters at $z\approx2$ have Lyman Continuum Escape Fractions $\approx50\%$
  The ionizing photon escape fraction (LyC $f_{\rm{esc}}$) of star-forming
galaxies is the single greatest unknown in the reionization budget. Stochastic
sightline effects prohibit the direct separation of LyC leakers from
non-leakers at significant redshift. Here we circumvent this uncertainty by
inferring $f_{\rm{esc}}$ with resolved (R>4000) LyA profiles from the X-SHOOTER
LyA survey at z=2 (XLS-z2). We select leakers ($f_{\rm{esc}}>20$%) and
non-leakers ($f_{\rm{esc}}<5$%) from a representative sample of $>0.2 L^{*}$
LyA emitters (LAEs). With median stacked spectra of these subsets covering
1000-8000 {\AA} (rest-frame) we investigate the conditions for LyC
$f_{\rm{esc}}$. We find the following differences between leakers vs.
non-leakers: (i) strong nebular CIV and HeII emission vs. non-detections, (ii)
O32~8.5 vs. ~3, (iii) Ha/Hb indicating no dust vs. E(B-V)~0.3, (iv) MgII
emission close to the systemic velocity vs. redshifted, optically thick MgII,
(v) LyA $f_{\rm{esc}}$ of ~50% vs. ~10%. The extreme EWs in leakers (O3+Hb~1100
{\AA}) constrain the characteristic timescale of LyC escape to ~3-10 Myr bursts
when short-lived stars with the hardest ionizing spectra shine. The defining
traits of leakers -- extremely ionizing stellar populations, low column
densities, a dust-free, high ionization state ISM -- occur simultaneously in
the $f_{\rm{esc}}>20\%$ stack, suggesting they are causally connected, and
motivating why indicators like O32 may suffice to constrain $f_{\rm{esc}}$ at
z>6 with JWST. The leakers comprise half our sample, have a median LyC
$f_{\rm{esc}}$~50%, and an ionising production efficiency
$\log({\xi_{\rm{ion}}/\rm{Hz\ erg^{-1}}})$~25.9. These results show LAEs -- the
type of galaxies rare at z=2, but that become the norm at higher redshift --
are highly efficient ionizers, with extreme $\xi_{\rm{ion}}$ and prolific
$f_{\rm{esc}}$ occurring in sync. (ABRIDGED)
