Cook the spaghetti in salted boiling water until just al dente—about 9 minutes. Drain it, but don't rinse; the starch helps the dressing cling. Spread it on a sheet pan to cool completely, which takes about 10 minutes.
While the pasta cools, toast the pine nuts in a dry skillet over medium heat for 3 minutes, stirring constantly. When you smell that nutty aroma, they're done. Don't walk away—I've burned a batch by getting distracted, and it only takes one mistake to learn this lesson.
Combine the 1/4 cup olive oil with minced garlic in a small bowl. Let it sit for 5 minutes so the garlic releases its flavor into the oil. This step is why the dressing tastes so much more sophisticated than store-bought versions.
Add the lemon zest to the garlic oil, then squeeze the lemon juice into the mixture. Whisk in the salt and black pepper. Taste it—you should detect bright lemon without any bitterness.
In a large bowl, toss the cooled spaghetti with the dressing until every strand glistens. This is what transforms cold lemon pasta crowd summer from ordinary to unforgettable. Add the shredded chicken, half the Parmesan, and three-quarters of the parsley.
Toss everything together gently so the pasta doesn't break. Top with remaining Parmesan, toasted pine nuts, and reserved parsley. Drizzle with 1 tablespoon olive oil for shine.
Let it rest for 15 minutes at room temperature before serving, or chill it up to 4 hours ahead. The flavors deepen as it sits, though I've noticed it tastes best eaten within 2 hours.