Cut the stem off the zucchinis, then use a mandolin to slice them into long thin ribbons. They should be about 1/8 inch thick, if they’re too thin they won’t stand up on the own, but if they’re too thick, they won’t be flexible enough to roll. Give each one a little sprinkle on both sides with salt and pepper.
Heat the grill to high, and do a quick sear of the zucchini. Since they are really thin, they should only be on the grill for about 30-60 seconds, just enough to take the raw edge off and give them some pretty little grill marks. Remove the grilled zucchini ribbons to a plate.
In a food processor, throw in the artichoke hearts, lemon zest, lemon juice, garlic, parsley, walnuts, and a big pinch of salt and pepper. Let the food processor run for a good 20 seconds, until everything is chopped up, and be sure to stop and scrape down the sides from time to time. With the food processor running, stream in the olive oil. Scrape the pesto out into a bowl, and stir in the grated parmesan. Taste it to check for proper seasoning.
Place a spoonful of artichoke pesto into each zucchini ribbon, and roll it up into a little roll. The artichoke pesto is sticky so it should stick together well. Serve and enjoy!