In the bowl of a food processor, add the parsley leaves, anchovy fillets, Parmesan, lemon juice and zest, green onions, chives and garlic. Season with salt and pepper. Pulse the mixture and slowly drizzle in the olive oil.
Remove the top from the food processor and unplug the machine. Replace the blade from the processor with the shredder blade and place the top back on the machine. Plug the machine back in and shred the lettuce leaves, 1 at a time, directly on top of the dressing.
Pour the lettuce and dressing into a large bowl and toss to combine. Garnish with additional Parmesan.