We prove that a "positive probability" subset of the boundary of "{uniformly expanding circle transformations}" consists of Kupka-Smale maps. More precisely, we construct an open class of $2$-parameter families of circle maps $(f_{a,\theta})_{a,\theta}$ such that for a positive Lebesgue measure subset of values of $a$, the family $(f_{a,\theta})_\theta$ leaves the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.