Sauté the Vegetables: In a Dutch oven or heavy-bottomed pot, heat 3 tablespoons of olive oil over medium heat until shimmering. Add the chopped yellow onion, shredded carrots, diced celery, minced garlic, kosher salt, and freshly ground black pepper. Cook, stirring occasionally, until the vegetables are softened and fragrant, approximately 5 minutes.
Add Broth and Chicken: Stir in 4 cups of low-sodium chicken broth, 2 cups of cooked shredded chicken, and 1 tablespoon of fresh thyme leaves. Bring the mixture to a boil, then reduce the heat to low and allow it to simmer for 10 minutes to allow the flavors to meld.
Incorporate Gnocchi and Spinach: Add 1 pound of refrigerated potato gnocchi, 1 (5-ounce) bag of baby spinach, and 1 1/2 cups of half-and-half to the pot. Cook for an additional 2 to 3 minutes, or until the gnocchi is tender and the spinach is wilted. Be careful not to overcook the gnocchi, as they should remain light and pillowy.
Serve Immediately: Ladle the soup into bowls and serve hot. For an enhanced dining experience, consider pairing it with a side of crusty bread or a light salad.