Operator learning is an increasingly popular technique for surrogate modeling that involves learning maps between function spaces. We present a multiscale kernel-based operator learning technique that leverages a multiscale frame approximation through compactly-supported kernels. The resulting operator is scale-aware upon generalization and scalable due to the sparsity of the multiscale frame approximation. Further, due to the connection between kernels and Gaussian processes (GPs), our multiscale operator also forms a multiscale GP, naturally admitting multiscale uncertainty estimates. We present numerical results on applications involving material fracture and fluid flow.