  We present projective Landau-Ginzburg models for the exceptional cominuscule
homogeneous spaces $\mathbb{OP}^2 = E_6^\mathrm{sc}/P_6$ and
$E_7^\mathrm{sc}/P_7$, known respectively as the Cayley plane and the
Freudenthal variety. These models are defined on the complement
$X^\vee_\mathrm{can}$ of an anti-canonical divisor of the "Langlands dual
homogeneous spaces" $\mathbb{X}^\vee = P^\vee\backslash G^\vee$ in terms of
generalized Pl\"ucker coordinates, analogous to the canonical models defined
for Grassmannians, quadrics and Lagrangian Grassmannians in arXiv:1307.1085,
arXiv:1404.4844, arXiv:1304.4958. We prove that these models for the
exceptional family are isomorphic to the Lie-theoretic mirror models defined in
arXiv:math/0511124 using a restriction to an algebraic torus, also known as the
Lusztig torus, as proven in arXiv:1912.09122. We also give a cluster structure
on $\mathbb{C}[\mathbb{X}^\vee]$, prove that the Pl\"ucker coordinates form a
Khovanskii basis for a valuation defined using the Lusztig torus, and compute
the Newton-Okounkov body associated to this valuation. Although we present our
methods for the exceptional types, they generalize immediately to the members
of other cominuscule families.
