Heat olive oil in a large pot, season chicken, and sear both sides until golden. Set aside.
Sauté onion and garlic in the same pot until translucent. Add zucchini and cook briefly.
Pour in chicken broth, shred chicken, and simmer together for 5 minutes.
In a food processor, blend basil, garlic, pine nuts, Parmesan, and olive oil into chunky pesto.
Stir in heavy cream, sun-dried tomatoes, and ¾ of the pesto into the soup. Adjust seasoning.
Add spinach or kale and cook until wilted. Finish with lemon juice if desired.