Reconstructing the center to edge spectra of stars in eclipsing binaries

Anish Amarsi showed (at GC Retreat 2018) that for the Sun, it is highly informative to analyze spectra from different parts of the disk, effectively spectra emerging under different angles from the photosphere; this is "straightfoward" for the Sun, as it is well resolved.

It looks like the math of doing this from multi-epoch spectra of eclipsing binaries is very straightfoward.

A very cursory check of the literature indicates that some efforts have been made in this direction (using H-alpha line profile to make a surface map etc..); but HWR has seen nothing comprehensive
to solve the problem:
Given a set of high-quality spectra of eclipsing binary stars, with sufficient phase coverage and known orbit geometry, is there a straightforward way to reconstruct rigorously the entire spectra of both stars, as a function of distance from the center; presuming cylindrical symmetry.

The answer to this question seems to be "yes, there is a simple way to do this"; see below.
This would afford spectra of the stars as function of the angle in the photosphere under which
these spectra emerge.

But before doing anything about this, HWR (who has chatted with Kareem about this for a minute)
would like to know/understand a few things form the stellar spectra experts (i.e. Anish/Karen):

• Is this actually an established approach, and HWR has just not found the relevant literature?
• How useful would it be to have spectra = f(cos theta) for different (mostly) MS stars, presumably across a wide range of Teff, and a modest range ( - 1.  <[Fe/H]<0.3 ) of metallicities (because bright eclipsing binaries are not that common..)?
• If useful, is it important this for testing models, or mostly for getting more accurate precise abundances? Or is this just a curiosity?
• If it is interesting in some Teff, FeH regime, what regime is the most interesting?

Having said that, here's a stab at the math of doing this "data driven"; in the naive version this seems very interesting.