Put the pesto sauce ingredients into a food processor or blender and process until almost smooth but still left with some chunk from the walnuts (if it is too thick, you can add a tablespoon or two of warm water).
In a large skillet, over medium heat, melt the butter.
Add the noodles and coat with the butter.
Add the sauce, about 1/2 cup at a time, and stir until the noodles are coated.
Top with the grated cheese and serve immediately, or at room temperature.