We investigate the asymptotic propagation speed of a Fisher-KPP equation on the plane with a Wentzell-type boundary condition imposed on the x-axis, which arises in a field-road model. Recently, this problem has been studied by Chen, He, and Wang (AMRA, 2023). Unlike their methods, in which fundamental solutions were mainly used, we utilize the theory of viscosity solutions for Hamilton-Jacobi equations to study the asymptotic expansion shape. With the aid of proper rescaling and WKB ansatz, a so-called phase function determines the asymptotic expansion shape. We prove that the phase function is the viscosity solution of a Hamilton-Jacobi variational inequality and can be identified by a value function of an optimal control problem. By solving this optimal control problem, we obtain the exact phase function formula and characterize the asymptotic expansion shape.