Heat 2 tablespoons olive oil in a large Dutch oven or large pot over medium heat until shimmering. Add 1 coarsely chopped yellow onion, 4 chopped medium carrots, 2 coarsely chopped medium celery stalks, 6 peeled garlic cloves, 1 ½ teaspoons kosher salt, and 1 teaspoon black pepper. Cook, stirring occasionally, until the vegetables are just beginning to brown, about 8 minutes.
Stir in 2 (32-ounce) cartons low-sodium chicken broth, 2 tablespoons soy sauce, 1 Parmesan cheese rind if using, 2 fresh thyme sprigs, and 2 dried bay leaves. Increase the heat to medium-high and bring to a boil. Reduce the heat to maintain a gentle simmer and cook until the vegetables are tender, 20 to 25 minutes.
Remove and discard the Parmesan rind, thyme stems, and bay leaves. Using a slotted spoon, transfer the vegetables to a blender. Blend until smooth.
Return the puréed vegetables to the pot. Bring the soup back to a boil over medium heat. Add 1 pound shredded chicken and 1 cup dried pastina pasta. Reduce the heat to maintain a simmer. Cook, stirring frequently and scraping the bottom of the pot, until the pasta is tender, about 6 minutes.
Turn off the heat. Stir in 1 tablespoon lemon juice. Taste and season with more kosher salt as needed. Garnish with grated Parmesan cheese and chopped fresh parsley leaves if desired.