We present an analysis of HI Lyman alpha 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 a 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 alpha line properties at sub-kpc scales. By combining all multiple images, we are able to observe complex structures in the Lyman a emission and uncover small (similar to 120 km s(-1) in Lyman alpha peak shift), but significant at >4 sigma, systematic variations in the shape of the Lyman a 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 alpha emission escapes preferentially from regions where the line profile has been less severely affected by scattering of Lyman alpha photons.