We consider the problem of designing static feedback gains subject to a priori structural constraints, which is a nonconvex problem in general. Previous work has focused on either characterizing special structures that result into convex formulations, or employing certain techniques to allow convex relaxations of the original problem. In this paper, by exploiting the underlying sparsity properties of the problem, and using chordal decomposition, we propose a scalable algorithm to obtain structured feedback gains to stabilize a large-scale system. We first extend the chordal decomposition theorem for positive semidefinite matrices to the case of matrices with block-chordal sparsity. Then, a block-diagonal Lyapunov matrix assumption is used to convert the design of structured feedback gains into a convex problem, which inherits the sparsity pattern of the original problem. Combining these two results, we propose a sequential design method to obtain structured feedback gains clique-by-clique over a clique tree of the block-chordal matrix, which only needs local information and helps ensure privacy of model data. Several illustrative examples demonstrate the efficiency and scalability of the proposed sequential design method.