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 461684 stars down to m_F555W_~29mag and m_F814W_~28mag, corresponding to the magnitude of an unreddened 1Myr pre-main-sequence star of ~0.09M_{sun}_ at the LMC distance. In this first paper we describe the observing strategy of MYSST and the data reduction procedure and present the photometric catalog. We identify multiple young stellar populations tracing the gaseous rim of N44's superbubble, together with various contaminants belonging to the LMC field population. We also determine the reddening properties from the slope of the elongated red clump (RC) feature by applying the machine-learning algorithm RANSAC, and we select a set of upper-main-sequence stars as primary probes to build an extinction map, deriving a relatively modest median extinction A_F555W_~0.77mag. The same procedure applied to the RC provides A_F555W_~0.68 mag.