Season the chicken. Place cubed chicken in a large mixing bowl, then season with 1 tsp Kosher salt, 1 tsp dried oregano, and ¼ tsp black pepper. Toss well.
Brown the chicken. Heat 2 Tbsp olive oil and 2 Tbsp butter in a large soup pot or dutch oven over medium-high heat. When hot, add the chicken and cook, stirring occasionally, until browned on all sides, about 4-5 minutes (the chicken will finish cooking in the soup). Use a slotted spoon to transfer to a plate.
Sauté the aromatics. To the pot, add 1 diced onion and 2 stalks of diced celery. Season with ½ tsp Kosher salt then cook, stirring occasionally, until translucent, about 3-5 minutes. Add ½ cup chopped sun-dried tomatoes, 4 cloves chopped garlic, and 2 Tbsp tomato paste. Stir, breaking up the paste with a spatula to help it dissolve.
Add the broth. Add 4 cups low-sodium chicken broth, 2 Tbsp pesto, and a Parmesan rind (if using). Bring the mixture to a boil, then reduce and simmer for 10-15 minutes.
Cook the gnocchi. Return the cooked chicken to the soup (along with any juices on the plate), then stir in 16oz dried potato gnocchi. Cook for 3-5 minutes, or until the gnocchi begin to float to the top and are tender.
Add cream. Stir in ½ cup heavy cream (or up to 1 cup if you want it really creamy), 5oz baby spinach, and ½ cup grated Parmesan cheese. Taste the soup, then adjust seasoning as needed, adding more Kosher salt or black pepper. Divide between bowls, then serve immediately with additional Parmesan cheese if wanted!