For most elements, polymict ureilite EET83309 shows no significant compositional difference from other ureilites, including ordinary ("monomict") ureilites. Polymict ureilites appear to be mixtures of a wide variety of ordinary ureilites, with little dilution by "foreign" extra-ureilitic materials. Thus, they apparently were mixed (i.e.• the ureilites in general formed) on a very small number of parent bodies. In one respect, polymict ureilites do stand out. Along with the only other polymict ureilite that has been analyzed for REE (Nilpena), EET83309 has much higher concentrations of light-middle REE than most ordinary ureilites. Despite these relative enrichments in LREE, polymict ureilites are nearly devoid ofbasaltic (Al-rich) material. A basaltic component should have formed along with (and presumably above) the ultramafic ureilites, in any closed-system differentiation ofan originally chondritic asteroid. This scarcity ofcomplementary basaltic materials may be an important clue to ureilite origins. We suggest that ureilites originated as paracumulates (mushy, cumulate-like, partial melt residues) deep within a primordiallyheated asteroid or asteroids. While still largely molten, the asteroid was severely disrupted, and most of its external basaltic portion was permanently blown away, by impact ofa large, C-rich projectile. This partially-disruptive impact tended to permeate the paracumulates with C-rich, noble-gas-rich, and IOO-rich magma derived mainly from shock-melting of the projectile. After reaccumulation and cooling, the resultant mixtures of cumulus mafic silicates with essentially "foreign" C-matrix became "monomict" ureilites. Further small impacts produced polymict ureilites as components of a newly-developed, basalt-poor megaregolith. The consistently moderate pyroxene/olivine ratios of the ureilites are as expected for partial melt residues, but not for cumulate (sensu stricto) rocks. The final projectile/target mixing ratio tended to be greatest among the more magnesian and pyroxene-rich portions of the paracumulate, because these portions were lowest in density, and thus concentrated toward the upper surface of the paracumulate layer. As a result, ureilites show correlations among C,~J70, and silicate-core mg. This model appears to reconcile many paradoxical aspects of ureilite composition (primitive, near-chondritic, except depleted in basalt, diverse~170) and petrography (igneous, cumulate-like).