Put the garlic, parsley, salt, and walnuts in a food processor. With the motor running, add the extra-virgin olive oil in a thin, steady stream. Once you have an even consistency, add the freshly grated Parmigiano cheese to the food processor, and puree for a few more seconds.
Pesto can be served fresh or it can be reserved in the refrigerator or the freezer. Do not add the nuts or the cheese before refrigerating or freezing as these two ingredients can spoil. Just process the garlic with the parsley, salt and oil, refrigerate or freeze in an airtight container, and add the processed nuts and cheese just before serving.