Add the basil, Parmesan, pine nuts and garlic to a food processor. Pulse a few times to break up the garlic and nuts. Add 1/4 cup olive oil and pulse a few more times, scraping down the sides as needed. With the motor running, stream in the remaining 1/4 cup olive oil and the lemon juice until a thick paste forms. Add salt and pepper to taste and pulse a few more times to combine.
Remove and store, refrigerated in an airtight container, for up to 2 days (with a thin layer of olive oil drizzled on top so that it does not turn brown).