Toss the chicken pieces in a mixing bowl with 1 tsp Kosher salt, the dried oregano, and the black pepper until evenly coated. The seasoning should cling to the chicken so every bite starts flavorful.

Heat the olive oil and butter in a large soup pot or Dutch oven over medium-high heat, then add the chicken and cook for 4-5 minutes, stirring now and then, until browned on all sides before transferring it to a plate with a slotted spoon. You’re looking for color on the outside, not fully cooked centers, because the chicken finishes later in the broth.

Add the onion and celery to the same pot, season with the remaining ½ tsp Kosher salt, and cook for 3-5 minutes until softened and translucent. Once the vegetables look glossy, stir in the sun-dried tomatoes, garlic, and tomato paste until the paste loosens and the pot smells fragrant and slightly sweet.

Pour in the chicken broth, then add the pesto and Parmesan rind, if using, and bring the soup to a boil before lowering the heat to a simmer for 10-15 minutes. The broth is ready when the flavors smell rounded and the liquid lightly coats the back of a spoon.

Return the browned chicken and any juices to the pot, then stir in the dried potato gnocchi and cook for 3-5 minutes. The gnocchi are done when they float to the top and feel tender instead of chalky.

Lower the heat and stir in ½ cup heavy cream, or up to 1 cup for a richer soup, along with the spinach and grated Parmesan, then taste and adjust with more Kosher salt or black pepper as needed. The soup should look creamy and glossy, and the spinach should just wilt into the hot broth before serving.