Using a vegetable peeler (preferably a Y-style peeler), remove the zest from the lemons in long, wide strips; try to remove only the colored portion of the peel, not the bitter white pith just underneath. You should have about ⅔ cup zest strips.

In a large pot, combine 2 quarts water, 1½ teaspoons salt, 1 teaspoon of sugar and half of the zest strips. Bring to a boil and to cook for 2 minutes, then remove and discard the zest. Add the spaghetti and cook until al dente. Reserve 1½ cups of the cooking water, then drain the pasta and return it to the pot.

Meanwhile, in a food processor, combine the remaining zest strips, the almonds, Parmesan, the remaining ½ teaspoon sugar and ¼ teaspoon each salt and pepper. Process until the mixture resembles coarse sand, 10 to 20 seconds. Add the oil and process just until the oil is incorporated (the mixture will not be smooth), about another 10 seconds; set aside until the pasta is ready.

To the spaghetti in the pot, add the pesto and ¾ cup of the reserved pasta water, then toss to combine; add more reserved pasta water as needed so the pesto coats the noodles. Toss in the chives. Taste and season with salt and pepper. Serve drizzled with additional oil and with additional grated Parmesan on the side.