Broyden’s method is a general method commonly used for nonlinear systems of equations when very little information is available about the problem. We develop an approach based on Broyden’s method for the structure appearing in nonlinear eigenvalue problems. Our approach is designed for problems where the evaluation of a matrix vector product is computationally expensive, essentially as expensive as solving the corresponding linear system of equations. We show how the structure of the Jacobian matrix can be incorporated into the algorithm to improve convergence. The algorithm exhibits local superlinear convergence for simple eigenvalues, and we characterize the convergence. We show how deflation can be integrated and combined such that the method can be used to compute several eigenvalues. A specific problem in machine tool milling, coupled with a PDE, is used to illustrate the approach. The simulations were carried out using the Julia programming language, and the simulation software is provided publicly for reproducibility.