Optional: Blanch the basil and tarragon. You may skip this step and add the herbs into the pesto fresh. The blanching process helps preserve the vibrant green color of the herbs, making your pesto a more vibrant green.
In a small pot, bring water to a boil and add herbs. Blanch for 15–25 seconds in boiling water, then immediately strain with a slotted spoon or fine mesh colander, and move to an ice bath to stop the cooking process. Remove from ice water and place on paper towels. Allow excess water to drain, then compile into a ball and gently squeeze out any extra water. By the end, the ball will be very compact and slightly damp, but not wet.
Pulse pine nuts in a food processor until separated into small pieces. Scrape down sides, then add garlic and olive oil. Pulse until ingredients are finely chopped in chunks, but not a paste texture—stop and scrape sides with spatula when needed.
Add the herbs, lemon juice, zest, salt, and pepper and pulse until incorporated—stopping every 15 seconds or so to scrape the sides.
Finally, add in the large cheese chunks and pulse until ingredients are finely chopped (but not pasty). Adjust seasoning with salt and pepper if needed, then enjoy on toast, roasted vegetables, pasta, and more!