Summary: It is well known that minimal surfaces over convex domains are always globally area-minimizing, which is not necessarily true for minimal surfaces over non-convex domains. Recently, M. Dorff, D. Halverson, and G. Lawlor proved that minimal surfaces over a bounded linearly accessible domain \(D\) of order \(\beta\) for some \(\beta \in (0, 1)\) must be globally area-minimizing, provided a certain geometric inequality is satisfied on the boundary of \(D\). In this article, we prove sufficient conditions for a sense-preserving harmonic function \(f=h+\overline{g}\) to be linearly accessible of order \(\beta\). Then, we provide a method to construct harmonic polynomials which maps the open unit disk \(\vert z \vert < 1\) onto a linearly accessible domain of order \(\beta\). Using these harmonic polynomials, we construct one parameter families of globally area-minimizing minimal surfaces over non-convex domains.