  The stellar mass-luminosity relation is poorly constrained by observations
for high mass stars. We describe our program to find eclipsing massive binaries
in the Magellanic Clouds using photometry of regions rich in massive stars, and
our spectroscopic follow-up to obtain radial velocities and orbits. Our
photometric campaign identified 48 early-type periodic variables, of which only
15 (31%) were found as part of the microlensing surveys. Spectroscopy is now
complete for 17 of these systems, and in this paper we present analysis of the
first two, LMC 172231 and ST2-28, simple detached systems of late-type O dwarfs
of relatively modest masses. Our orbit analysis yields very precise masses (2%)
and we use tomography to separate the components and determine effective
temperatures by model fitting, necessary for determining accurate (0.05-0.07
dex) bolometric luminosities in combination with the light-curve analysis. Our
approach allows more precise comparisons with evolutionary theory than
previously possible. To our considerable surprise, we find a small, but
significant, systematic discrepancy: all of the stars are slightly
under-massive, by typically 11% (or over-luminous by 0.2 dex) compared to that
predicted by the evolutionary models. We examine our approach for systematic
problems, but find no satisfactory explanation. The discrepancy is in the same
sense as the long-discussed and elusive discrepancy between the masses measured
from stellar atmosphere analysis with the stellar evolutionary models, and
might suggest that either increased rotation or convective overshooting is
needed in the models. Additional systems will be discussed in future papers of
this series, and will hopefully confirm or refute this trend.
