Mesoscale eddies, generated by lateral gradients in salinity and temperature in the Arctic marginal ice zone (MIZ), are known to modulate the melting of sea ice in this region. Yet, it remains unclear if eddies also modify sea ice growth during the freezing season. Here, we use a set of idealized simulations to explore the sea ice growth above an eddying ocean. In the presence of eddies, mixing of the surface temperature and salinity fields induce heterogeneity in the heat and salt fluxes at the ice-ocean interface, ultimately imprinting heterogeneity on the sea ice thickness. A stronger eddy field imprints more heterogeneity in the sea ice thickness. More heterogeneity in the sea ice pack would likely impact the current and future evolution of the sea ice conditions in the Arctic, where a rapid transition towards an open-ocean regime is ongoing.