We report a discovery of six massive galaxies with both extremely large Ly alpha equivalent widths (EWs) and evolved stellar populations at z ~ 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 12 optical intermediate-band data taken with Subaru/Suprime-Cam in the COSMOS field. Based on the spectral energy distribution fitting analysis for these LAEs, it is found that these MAESTLOs have (1) large rest-frame EWs of EW sub(0) (Ly alpha ) ~ 100-300 Angstrom, (2) Mlow * ~ 10 super(10.5)-10 super(11.1) M sub(middot in circle), and (3) relatively low specific star formation rates of SFR/Mlow * ~ 0.03-1 Gyr super(-1). Three of the six MAESTLOs have extended Ly alpha emission with a radius of several kiloparsecs, 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 active galactic nuclei, the observed extended Ly alpha emission is likely to be caused by a 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.