Preheat the oven to 400 degrees
Cut off the woody ends of the asparagus, Save the ends for stock, soup, or compost.
Place stalks on a sheet pan. Drizzle on some olive oil, then sprinkle on kosher salt to taste.
Take a head of garlic wrap it in foil and place it in the oven along with the asparagus.
Roast for about 15-20 minutes until asparagus is brown and has softened. Check half way through to set if the tips are starting to brown and cover then with aluminum foil. Check if the garlic has softened, leave it oven for a little longer if needed.
Allow to cool for a few minutes before cutting off the tips. Set them aside. Then slice up the stems into bite sized pieces.
Place in your food processor: the roasted garlic, asparagus stem pieces, and Parmesan cheese, and the juice of a lemon. Process to combine, making sure to scrape down the sides of the bowl at least once.
With the machine running drizzle in about ¼ cup of olive oil to start. Add more oil to get to the right consistency you desire. Drizzle slowly and keep checking. Add additional salt if needed.
Remove the pesto from the food processor and fold in the stems. Serve over the cooked pasta. Garnish with more cheese.