Chiral symmetry is always broken in cold, dense matter, by chiral condensation at low densities and by diquark condensation at high density. We construct here, within a schematic Nambu-JonaLasinio (NJL) model, the corresponding generalized Nambu-Goldstone pion, πG. As we show, the πG mode naturally emerges as a linear combination of the qq vacuum pion π and the qq diquarkcondensate pionπ, with q the quark field, and continuously evolves with increasing density from being π-like in the vacuum toπ-like in the high density diquark pairing phase. We calculate the density-dependent mass, decay constant, and coupling to quarks of the πG, and derive a generalized Gell-Mann-Oakes-Renner (GMOR) relation in the presence of a finite bare quark mass mq. We briefly discuss the implications of the results to possible Bose condensation of πG in more realistic models.