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Comparison of echelle spectra to synthetic models has become a computational statistics challenge, with over ten thousand individual spectral lines affecting a typical cool star echelle spectrum. Telluric artifacts, imperfect line lists, inexact continuum placement, and inflexible models frustrate the scientific promise of these information-rich datasets. Here we debut an interpretable machine-learning framework "blasé" that addresses these and other challenges. The semi-empirical approach can be viewed as "transfer learning" -- first pre-training models on noise-free precomputed synthetic spectral models, then learning the corrections to line depths and widths from whole-spectrum fitting to an observed spectrum. The auto-differentiable model employs back-propagation, the fundamental algorithm empowering modern Deep Learning and Neural Networks. Here, however, the 40,000+ parameters symbolize physically interpretable line profile properties such as amplitude, width, location, and shape, plus radial velocity and rotational broadening. This hybrid data-/model- driven framework allows joint modeling of stellar and telluric lines simultaneously, a potentially transformative step forwards for mitigating the deleterious telluric contamination in the near-infrared. The blasé approach acts as both a deconvolution tool and semi-empirical model. The general purpose scaffolding may be extensible to many scientific applications, including precision radial velocities, Doppler imaging, chemical abundances, and remote sensing. Its sparse-matrix architecture and GPU-acceleration make blasé fast. The open-source PyTorch-based code includes tutorials, Application Programming Interface (API) documentation, and more. We show how the tool fits into the existing Python spectroscopy ecosystem, demonstrate a range of astrophysical applications, and discuss limitations and future extensions.