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The Fermi mission opened a new era for pulsar astronomy, detecting gammaraypulsations from more than 60 pulsars, 30% of which are not seen at radiowavelengths. One of the most interesting Fermi pulsars is PSR J0357+32, aradioquiet source standing out as the gammaray pulsar with the smallestspindown luminosity (Erotvirgul1e33 erg/s) known so far. We observed PSR J0357+32with Chandra and we detected its Xray counterpart, but we could not constrainits emission properties. Chandra data unveiled the existence of a huge (9 arcminlong) extended feature apparently protruding from the pulsar, with a peculiarphenomenology. We ask for a deep observation with XMM in order to assess thepulsar emission properties as well as the nature of the extended feature. truncated!, Please see actual data for full text [truncated!, Please see actual data for full text]