We develop an efficient and adaptive sparse reconstruction approach for the recovery of spectral images from the measurements of a photon-sieve spectral imager (PSSI). PSSI is a computational imaging technique that enables higher resolution than conventional spectral imagers. Each measurement in PSSI is a superposition of the blurred spectral images; hence, the inverse problem can be viewed as a type of multi-frame deconvolution problem involving multiple objects. The transform learning-based approach reconstructs the spectral images from these superimposed measurements while simultaneously learning a sparsifying transform. This is performed using a block coordinate descent algorithm with efficient update steps. The performance is illustrated for a variety of measurement settings in solar spectral imaging. Compared to approaches with fixed sparsifying transforms, the approach is capable of efficiently reconstructing spectral images with improved reconstruction quality.