Combine radish leaves, basil leaves, garlic, toasted walnuts, and one tablespoon of olive oil into a food processor. Pulse until ingredients are starting to form a paste. Stop to scrape down radish and basil leaves whenever necessary.
Add the parmesan cheese, salt and pepper. Pulse a few times to combine everything. Then, slowly drizzle in the additional olive oil as you turn the processor to a steady blending mode. Blend until creamy, or to your preferred texture.
Add additional salt to taste, and add additional olive oil to achieve a creamier texture if you like. Refrigerate pesto for up to one week, or freeze for up to three months.