  We provide minimal constructions of meanders with particular combinatorics.
Using these meanders, we give minimal constructions of hyperelliptic pillowcase
covers with a single horizontal cylinder and simultaneously a single vertical
cylinder so that one or both of the core curves are separating curves on the
underlying surface. In the case where both of the core curves are separating,
we use these surfaces in a construction of Aougab-Taylor in order to prove that
for any hyperelliptic connected component of the moduli space of quadratic
differentials with no poles there exist ratio-optimising pseudo-Anosovs lying
arbitrarily deep in the Johnson filtration and stabilising the Teichm\"uller
disk of a quadratic differential lying in this connected component.
