In 2020, we introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. In particular, digraphs that have a restricted numerical range of a single point, horizontal line segment, and vertical line segment were characterized as k-imploding stars, directed joins of bidirectional digraphs, and regular tournaments, respectively. We now extend this work by investigating digraphs whose restricted numerical range forms a convex polygon in the complex plane. We provide computational methods for identifying these polygonal digraphs, and show that these digraphs can be broken into three disjoint classes that are closed under digraph complement: normal, restricted-normal, and pseudo-normal digraphs. We prove sufficient conditions for normal digraphs and show that the directed join of two normal digraphs results in a restricted-normal digraph. Also, we prove that directed joins are the only restricted-normal digraphs when the order is square-free or twice a square-free number. Finally, we provide a construction for restricted-normal digraphs that are not directed joins for all orders that are neither square-free nor twice a square-free number.