We propose the multi-semester Exo-EASY survey to obtain Elemental Abundances via SpectroscopY of Exoplanet-hosting cool dwarfs. Observations of stellar atmospheres reveal the primitive materials from which planetary systems form. If measured with sufficient precision, stellar abundances can place strong constraints on planet properties including planet formation, and bulk composition, and atmospheric compositions. Historically elemental abundance studies often focus on FGK stars while avoiding cooler dwarfs, but cool dwarfs host host more planets per star and are intrinsically more common. Exo-EASY will use IGRINS and IGRINS2 over six semesters to measure precise abundances of 9-15 elements in 170 cool dwarf stars via high-resolution NIR spectroscopy. This sample will include: (1) JWST exoplanet host stars with planets being observed in Cycles 1--5; (2) stars hosting small, rocky exoplanets with precisely-measured masses and radii; (3) stars hosting multi-planet systems; and (4) a hot/cool (M/FGK) binary sample for calibration of abundances. Our program will (1) enable precise interior and atmospheric modeling to reveal the overall makeup of other worlds; (2) allow better interpretation of ongoing compositional measurements from JWST and interior constraints from RV+transit surveys; and (3) reveal connections between stellar abundances, planet properties, system architecture, and planet formation.