The relative abundances of singly deuterated methanol isotopologues, [CH2DOH]/[CH3OD], in star-forming regions deviate from the statistically expected ratio of 3. In Orion KL, the nearest high-mass star-forming region to Earth, the singly deuterated methanol ratio is about 1, and the cause for this observation has been explored through theory for nearly three decades. We present high-angular resolution observations of Orion KL using the Atacama Large Millimeter/submillimeter Array to map small-scale changes in CH3OD column density across the nebula, which provide a new avenue to examine the deuterium chemistry during star and planet formation. By considering how CH3OD column densities vary with temperature, we find evidence of chemical processes that can significantly alter the observed gas-phase column densities. The astronomical data are compared with existing theoretical work and support D-H exchange between CH3OH and heavy water (i.e., HDO and D2O) at methanol's hydroxyl site in the icy mantles of dust grains. The enhanced CH3OD column densities are localized to the Hot Core-SW region, a pattern that may be linked to the coupled evolution of ice mantle chemistry and star formation in giant molecular clouds. This work provides new perspectives on deuterated methanol chemistry in Orion KL and informs considerations that may guide future theoretical, experimental, and observational work.