Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a scheme whose small site is homeomorphic to the small site associated to $S^1$ (considered as a topological space)? By a homeomorphism of sites I mean a continuous equivalence of the underlying categories (with a continuous inverse).

See here for some remarks about etale morphisms (I do not think that etale site would work, however).