  A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth
subvariety of the flag variety determined by a square matrix $S$ with distinct
eigenvalues and a Hessenberg function $h$. The cohomology ring
$H^*(\mathrm{Hess}(S,h))$ is independent of the choice of $S$ and is not
explicitly described except for a few cases. In this paper, we characterize the
Hessenberg function $h$ such that $H^*(\mathrm{Hess}(S,h))$ is generated in
degree two as a ring. It turns out that such $h$ is what is called a (double)
lollipop.
