Two-dimensional cadmium selenide nanoplatelets (NPLs) exhibit large absorption cross sections and homogeneously broadened band-edge transitions that offer utility in wide-ranging optoelectronic applications. Here, we examine the temperature-dependence of amplified spontaneous emission (ASE) in 4- and 5-monolayer thick NPLs and show that the threshold for close-packed (neat) films decreases with decreasing temperature by a factor of 2-10 relative to ambient temperature owing to extrinsic (trapping) and intrinsic (phonon-derived line width) factors. Interestingly, for pump intensities that exceed the ASE threshold, we find development of intense emission to lower energy in particular provided that the film temperature is ≤200 K. For NPLs diluted in an inert polymer, both biexcitonic ASE and low-energy emission are suppressed, suggesting that described neat-film observables rely upon high chromophore density and rapid, collective processes. Transient emission spectra reveal ultrafast red-shifting with the time of the lower energy emission. Taken together, these findings indicate a previously unreported process of amplified stimulated emission from polyexciton states that is consistent with quantum droplets and constitutes a form of exciton condensate. For studied samples, quantum droplets form provided that roughly 17 meV or less of thermal energy is available, which we hypothesize relates to polyexciton binding energy. Polyexciton ASE can produce pump-fluence-tunable red-shifted ASE even 120 meV lower in energy than biexciton ASE. Our findings convey the importance of biexciton and polyexciton populations in nanoplatelets and show that quantum droplets can exhibit light amplification at significantly lower photon energies than biexcitonic ASE.