JWST observations have recently begun delivering the first samples of Lyα velocity profile measurements at z > 6, opening a new window into the reionization process. Interpretation of z ≳ 6 line profiles is currently stunted by limitations in our knowledge of the intrinsic Lyα profile (before encountering the intergalactic medium (IGM)) of the galaxies that are common at z ≳ 6. To overcome this shortcoming, we have obtained resolved (R ∼ 3900) Lyα spectroscopy of 42 galaxies at z = 2.1-3.4 with similar properties as are seen at z > 6. We quantify a variety of Lyα profile statistics as a function of [O iii]+Hβ equivalent width (EW). Our spectra reveal a new population of z ≃ 2-3 galaxies with large [O iii]+Hβ EWs (>1200 Å) and a large fraction of Lyα flux emerging near the systemic redshift (peak velocity ≃0 km s−1). These spectra indicate that low-density neutral hydrogen channels are able to form in a subset of low-mass galaxies (≲1 × 108 M ⊙) that experience a burst of star formation (sSFR > 100 Gyr−1). Other extreme [O iii] emitters show weaker Lyα that is shifted to higher velocities (≃240 km s−1) with little emission near the line center. We investigate the impact the IGM is likely to have on these intrinsic line profiles in the reionization era, finding that the centrally peaked Lyα emitters should be strongly attenuated at z ≳ 5. We show that these line profiles are particularly sensitive to the impact of resonant scattering from infalling IGM and can be strongly attenuated even when the IGM is highly ionized at z ≃ 5. We compare these expectations against a new database of z ≳ 6.5 galaxies with robust velocity profiles measured with JWST/NIRSpec.