Coherent neutrino-nucleon scattering offers a unique approach in the search for physics beyond the standard model. When used in conjunction with monoenergetic neutrino sources, the technique can be sensitive to the existence of light sterile neutrinos. The ability to utilize such reactions has been limited in the past due to the extremely low-energy threshold (10--50 eV) needed for detection. In this paper, we discuss an optimization of cryogenic solid-state bolometers that enables reaching extremely low kinetic energy thresholds. We investigate the sensitivity of an array of such detectors to neutrino oscillations to sterile states. A recent analysis of available reactor data appears to favor the existence of such a sterile neutrino with a mass splitting of $|\ensuremath{\Delta}{m}_{\mathrm{sterile}}{|}^{2}\ensuremath{\ge}1.5\text{ }\text{ }{\mathrm{eV}}^{2}$ and mixing strength of ${sin}^{2}2{\ensuremath{\theta}}_{\mathrm{sterile}}=0.17\ifmmode\pm\else\textpm\fi{}0.08$ at 95% confidence level. An array of such low-threshold detectors would be able to make a definitive statement as to the validity of the interpretation.