  In order to better understand the role of high-mass stellar feedback in
regulating star formation in giant molecular clouds, we carried out a Hubble
Space Telescope (HST) Treasury Program "Measuring Young Stars in Space and
Time" (MYSST) targeting the star-forming complex N44 in the Large Magellanic
Cloud (LMC). Using the F555W and F814W broadband filters of both the ACS and
WFC3/UVIS, we built a photometric catalog of 461,684 stars down to
$m_\mathrm{F555W} \simeq 29$ mag and $m_\mathrm{F814W} \simeq 28$ mag,
corresponding to the magnitude of an unreddened 1 Myr pre-main-sequence star of
$\approx0.09$ $M_\odot$ at the LMC distance. In this first paper we describe
the observing strategy of MYSST, the data reduction procedure, and present the
photometric catalog. We identify multiple young stellar populations tracing the
gaseous rim of N44's super bubble, together with various contaminants belonging
to the LMC field population. We also determine the reddening properties from
the slope of the elongated red clump feature by applying the machine learning
algorithm RANSAC, and we select a set of Upper Main Sequence (UMS) stars as
primary probes to build an extinction map, deriving a relatively modest median
extinction $A_{\mathrm{F555W}}\simeq0.77$ mag. The same procedure applied to
the red clump provides $A_{\mathrm{F555W}}\simeq 0.68$ mag.
