ABSTRACT We present an analysis of ${\rm H\,\rm{\small {I}}}$ Lyman $α$ emission in deep VLT/MUSE observations of two highly magnified and extended galaxies at $z=3.5$ and 4.03, including a newly discovered, almost complete Einstein ring. While these Lyman $α$ haloes are intrinsically similar to the ones typically seen in other MUSE deep fields, the benefits of gravitational lensing allow us to construct exceptionally detailed maps of Lyman $α$ line properties at sub-kpc scales. By combining all multiple images, we are able to observe complex structures in the Lyman $α$ emission and uncover small ($∼120$ km s−1 in Lyman $α$ peak shift), but significant at $ \gt $4 $σ$, systematic variations in the shape of the Lyman $α$ line profile within each halo. Indeed, we observe a global trend for the line peak shift to become redder at large radii, together with a strong correlation between the peak wavelength and line width. This systematic intrahalo variation is markedly similar to the object-to-object variations obtained from the integrated properties of recent large samples. Regions of high surface brightness correspond to relatively small line shifts, which could indicate that Lyman $α$ emission escapes preferentially from regions where the line profile has been less severely affected by scattering of Lyman $α$ photons.