Add enough water to cover the bottom of a saucepan and add the broccoli. Cover the pot and cook over medium heat until the broccoli begins to soften - about 10 minutes.
Drain the broccoli and reserve some of the water.
Place the broccoli, basil, garlic, and walnuts in the food processor bowl.
Process the pesto mixture until it is almost pureed.
Remove the top of the food processor, scrape down the sides add the shredded cheese.
Process the pesto once again adding the oil through the top of the food processor.
Adjust the consistency using a little bit of oil or the reserved water from the broccoli.
Notes
Store leftovers in the refrigerator for up to four days in an airtight container.Freeze broccoli pesto sauce for up to three months.See blog post for serving suggestions and tips.