One half of mirror symmetry for Fano varieties is typically stated as a relation between the symplectic geometry of a Fano variety Y and the complex geometry of a Landau-Ginzburg model, realized as a pair (X,W) with X a quasi-projective variety and W a regular function on X. The pair (X,W) itself is expected to reflect a pair on the Fano side, namely a decomposition of Y into a disjoint union of an affine log Calabi-Yau and an anticanonical divisor D, thought of as mirror to W. We will discuss recent work which shows how the intrinsic mirror construction of Gross and Siebert naturally produce potential LG models assuming milder conditions on the singularities of D than typically required for the intrinsic mirror construction. In particular, we show that classical periods of this LG model recover the quantum periods of Y. In the setting when Y\D is an affine cluster variety, we will describe how these LG models naturally give rise to Laurent polynomial mirrors and encode certain toric degenerations of Y. As an example, we consider Y = Gr(k,n), D a particular choice of anticanonical divisor with affine cluster variety complement and give an explicit description of the intrinsic LG model in terms of Plücker coordinates on Gr(n-k,n), recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams.