The ionizing photon escape fraction [Lyman continuum (LyC) fesc] 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 redshifts. Here we circumvent this uncertainty by inferring fesc using resolved (R > 4000) Lyman α (Lyα) profiles from the X-SHOOTER Lyα survey at z = 2 (XLS-z2). With empirically motivated criteria, we use Lyα profiles to select leakers ($f_{\mathrm{ esc}} > 20{{\ \rm per\ cent}}$) and non-leakers ($f_{\mathrm{ esc}} < 5{{\ \rm per\ cent}}$) from a representative sample of >0.2L* Lyman α emitters (LAEs). We use median stacked spectra of these subsets over λrest ≍ 1000-8000 Å to investigate the conditions for LyC fesc. Our stacks show similar mass, metallicity, MUV, and βUV. We find the following differences between leakers versus non-leakers: (i) strong nebular C IV and He II emission versus non-detections; (ii) [O III]/[O II] ≍ 8.5 versus ≍3; (iii) Hα/Hβ indicating no dust versus E(B - V) ≍ 0.3; (iv) Mg II emission close to the systemic velocity versus redshifted, optically thick Mg II; and (v) Lyα fesc of ${\approx} 50{{\ \rm per\ cent}}$ versus ${\approx} 10{{\ \rm per\ cent}}$. The extreme equivalent widths (EWs) in leakers ([O III]+$\mathrm{ H}\beta \approx 1100$ Å rest frame) constrain the characteristic time-scale 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 interstellar medium (ISM) - occur simultaneously in the $f_{\rm esc} > 20{{\ \rm per\ cent}}$ stack, suggesting they are causally connected, and motivating why indicators like [O III]/[O II] may suffice to constrain fesc at z > 6 with the James Webb Space Telescope (JWST). The leakers comprise half of our sample, have a median LyC$f_{\rm esc} \approx 50{{\ \rm per\ cent}}$ (conservative range: $20\!-\!55{{\ \rm per\ cent}}$), and an ionizing production efficiency $\log ({\xi _{\rm {ion}}/\rm {Hz\ erg^{-1}}})\approx 25.9$ (conservative range: 25.7-25.9). 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 ξion and prolific fesc occurring in sync.