Place the basil in the bowl of a food processor. (It’s ok to use some of the stems.) Top with the cheese, seeds, lemon juice, garlic, salt, and pepper. Pulse a few times to coarsely chop, and then process in a few longer bursts; you want the mixture to be evenly chopped but still have some texture.
Scrape down the sides, and then with the machine running, add the oil in a fine stream through the feed tube. Process until pesto is mostly smooth. (I like to keep just a hint of texture.)
Use immediately or transfer the mixture to an airtight container and refrigerate. The top layer will brown slightly over time but the pesto will still taste good. (See tip.)
To store in the freezer, transfer to freezer-safe containers***, add a thin layer of olive oil to cover the surface, put on the lids, and then label with the date and freeze. The frozen pesto will keep for about a year.