We construct an $mathcal{N} = 2$ supersymmetric sigma model on the cotangent bundle over the Hermitian symmetric space $E_7 / (E_6 × U(1))$ in the projective superspace formalism, which is a manifest $mathcal{N} = 2$ off-shell superfield formulation in four-dimensional spacetime. To obtain this model we elaborate on the results developed in arXiv:0811.0218 and present a new closed formula for the cotangent bundle action, which is valid for all Hermitian symmetric spaces. We show that the structure of the cotangent bundle action is closely related to the analytic structure of the Kähler potential with respect to a uniform rescaling of coordinates.