  In every connected component of every stratum of Abelian differentials, we
construct square-tiled surfaces with one vertical and one horizontal cylinder.
We show that for all but the hyperelliptic components this can be achieved in
the minimum number of squares necessary for a square-tiled surface in that
stratum. For the hyperelliptic components, we show that the number of squares
required is strictly greater and construct surfaces realising these bounds.
Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms
optimising the ratio of Teichm\"uller to curve graph translation length are, in
a reasonable sense, ubiquitous in the connected components of strata of Abelian
differentials. Finally, we present a further application to filling pairs on
punctured surfaces by constructing filling pairs whose algebraic and geometric
intersection numbers are equal.
