The mixed layer is the uppermost layer of the ocean, driven by atmospheric fluxes. It follows a strong seasonal cycle, deepening in winter due to buoyancy loss, shoaling very close to the surface with summer restratification. Recently global and regional studies showed a mixed layer depth (MLD) modulation by mesoscale eddies with the seasonal cycle. In winter, MLD tends to be deeper inside anticyclonic eddies and shallower inside cyclonic ones. Several studies proposed a scaling law with eddy sea surface height deviation. However they were done globally or regionally with eddy composites mostly representative of surface-intensified structures and using monthly averaged climatologies as reference. The Mediterranean sea contains a wide variety of mesoscale eddies, with the specific presence of several large anticyclones living up to 4 years, in particular in the Eastern basin. These anticyclones were surveyed over the past decade with numerous Argo floats deployments. Several floats remained trapped inside anticyclones for months and enabled to record 16 continuous MLD time series inside 13 long-lived anticyclones at a fine temporal scale on the order of the week. MLD evolution at anticyclone cores reveals a stronger winter deepening, reaching sometimes deeper than 300 m, compared to always less than 100 m in the neighboring background. MLD evolution also does not coincide inside- and outside-eddy, starting to restratify outside of eddies, while it keeps deepening and cooling MLD at anticyclone core for a longer time. We then bring to light a restratification delay of one month on average between the anticyclones and their background, sometimes reaching more than 2 months. Extreme MLD anomalies of up to 330 m that would be smoothed in composite analyses can then be observed when the winter mixed layer connects with a preexisting subsurface anticyclonic core, greatly accelerating mixed layer deepening. On the opposite, the winter MLD sometimes does not achieve such connection but homogenizes a second subsurface layer, then forming a double-core anticyclone with spring restratification. Formation of several double-core anticyclones in the Eastern Mediterranean is accurately described in time. MLD restratification delay and connection with preexisting subsurface anomalies appear to be determinant in MLD modulation by mesoscale eddy and highlights the importance of interaction with eddy vertical structure.