Use a potato peeler to remove the rind off of the lemons. Be careful to only take the yellow part of the outer skin (zest) and a small amount of the white pith, which is fairly bitter. Juice enough lemons to yield 5 tablespoons of lemon juice (about 2 medium lemons). Set the lemon rinds and juice aside.
Add water, ¼ of the lemon rinds, and 1 tablespoon of lemon juice to a large pot and bring the mixture to a boil. Add the linguine and cook according to package directions for al dente pasta (about 7-8 minutes). Reserve 1 cup of pasta water. Remove the lemon peels. Drain the pasta and set aside. *adding rinds and juice helps infuse the pasta with extra lemon flavor, but you can omit this step if preferred.
In a food processor, combine the remaining lemon rinds, garlic, nuts, cheese, sugar, salt and pepper. Pulse until the mixture resembles coarse sand. Add the olive oil and remaining 4 tablespoons of lemon juice and pulse until just incorporated (about 20 seconds). The mixture will look fairly creamy. Set aside.
Return the cooked pasta to the pot. Add ⅓ of the pesto and ¼ cup of the pasta water to start. You'll want the mixture to be runny enough to resemble a sauce, but thick enough to cling to the noodles. Taste and add more pesto, pasta water, and/or salt and pepper according to your tastes. *We generally use most of the pesto and about ⅔ to ¾ cups reserved pasta water.
Divide among plates and garnish with additional cheese, pine nuts, and/or fresh herbs. Serve with shrimp, chicken, or another protein to make it a complete meal.