Water ice has been delivered to the lunar poles from different sources over billions of years, but this accumulation was punctuated by large impacts that excavated dry regolith from depth and emplaced it in layers over the poles. Here, we model the resulting stratigraphies of ice and ejecta deposits in the lunar polar regions. Large polar craters were age dated, and their ejecta distributions calculated with standard scaling relations. We then created a Monte Carlo model for ice deposition and ejecta emplacement. Typical model runs showed that deposits in older cold traps (>4 Ga) are divided into two zones: buried ice-rich gigaton deposits and younger more gardened mantles. The latter are consistent with small crater morphometry measurements, but the existence of substantial ice buried at great depths is more difficult to confirm. Rare outlier model runs included Mercury-like cases with significant deposition events in recent history (<200 Ma). Plain Language Summary The polar regions of Earth's Moon have topographic depressions that are never directly exposed to the Sun, so they are cold enough for deposits of ice to exist. Water can get into these regions by water-bearing asteroids colliding with the Moon, or from lunar volcanoes erupting gases that travel to the poles. At the same time, large impact craters that form at the poles eject an enormous amount of soil and rock that could bury existing ice. It is not well understood how these two processes work together to build up deposits that may have alternating layers of ice-rich and ice-poor soil. In this study, we used computer simulations to predict what these layered deposits may look like. We found it is likely that large amounts of relatively pure ice are buried at depth in the oldest deposits, covered with thinner layers hosting less ice. Impact cratering has been the dominant process affecting the lunar poles, but the effects of large polar craters on nearby ice deposits have not been previously addressed. Impact effects have been considered for micrometeoroids (e.g.,