We report a discovery of 6 massive galaxies with both extremely large Ly \alpha equivalent width and evolved stellar population at z \sim 3 . These MAssive Extremely STrong Ly \alpha emitting Objects ( MAESTLOs ) have been discovered in our large-volume systematic survey for strong Ly \alpha emitters ( LAEs ) with twelve optical intermediate-band data taken with Subaru/Suprime-Cam in the COSMOS field . Based on the SED fitting analysis for these LAEs , it is found that these MAESTLOs have ( 1 ) large rest-frame equivalent width of EW _ { 0 } ( \mathrm { Ly } \alpha ) \sim 100 –300 Å , ( 2 ) M _ { \star } \sim 10 ^ { 10.5 } – 10 ^ { 11.1 } ~ { } M _ { \odot } , and ( 3 ) relatively low specific star formation rates of SFR / M _ { \star } \sim 0.03 – 1 ~ { } \mathrm { Gyr } ^ { -1 } . Three of the 6 MAESTLOs have extended Ly \alpha emission with a radius of several kpc although they show very compact morphology in the HST/ACS images , which correspond to the rest-frame UV continuum . Since the MAESTLOs do not show any evidence for AGNs , the observed extended Ly \alpha emission is likely to be caused by star formation process including the superwind activity . We suggest that this new class of LAEs , MAESTLOs , provides a missing link from star-forming to passively evolving galaxies at the peak era of the cosmic star-formation history .