Elliptic Grid Generation (EGG), using the Winslow generator, defines a map between a simple computational region and a potentially complicated physical region. It can be used numerically to create meshes for discretizing equations directly on the physical domain or indirectly on the computational domain by way of the transforming map. EGG allows complete specification of the boundary, and it guarantees a one-to-one and onto transformation when the computational region is convex. A new fully variational approach is developed for solving the Winslow equations that enables accurate discretization and fast solution methods. The EGG equations are converted to a first-order system that is then linearized via Newton's method. First-order system least squares (FOSLS) is used to formulate and discretize the Newton step, and the resulting matrix equation is solved using algebraic multigrid (AMG). The approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse-level computation. Theoretical and numerical results confirm the usual full multigrid efficiency: accuracy comparable to the finest-level discretization is achieved at a cost proportional to the number of finest-level degrees of freedom.