Make the pesto. Fit a large food processor with a standard blade. Add artichokes and garlic, pulse several times until the artichokes are roughly chopped. Add basil, lemon zest, lemon juice and walnuts. Pulse 4-5 times. Scrape down the sides of the bowl. Add pepper and then, with the motor running slowly pour the olive oil down the chute and continue processing until fully blended.
Cook the pasta. Cook according to the package directions. When the pasta is done cooking drain it, reserving at least a ⅓ cup of the pasta water.
Cook the shrimp. Heat a large skillet over medium heat. While the skillet is heating toss the shrimp with olive oil and garlic. Add shrimp to the skillet and cook for 2 minutes. Using a pair of tongs, turn the shrimp and cook for an additional 1-2 minutes until shrimp are pink and cooked through.
Finish the dish. Turn the heat off and add the pasta, pesto, and ¼ cup of the pasta water to the pan. Carefully toss to coat the linguine. Add additional pasta water, a small amount at a time, as needed. Serve immediately, topping with grated parmesan if desired.