Heat 3 tablespoons olive oil in a Dutch oven or heavy-bottomed pot over medium heat until shimmering. Add chopped large yellow onion, 1 cup shredded carrots, 2 stalks diced celery, 4 cloves minced garlic, 1 ½ teaspoons kosher salt, and ½ teaspoon freshly ground black pepper, and cook, stirring occasionally until softened, about 5 minutes.
Stir in 4 cups low-sodium chicken broth, 2 cups cooked shredded chicken, and 1 tablespoon fresh thyme leaves, and bring to a boil. Reduce the heat and simmer for 10 minutes. Add 1 pound refrigerated potato gnocchi, 1 (5-ounce) bag baby spinach, and 1 ½ cups half-and-half, and cook until the gnocchi is cooked through and the spinach is wilted, 2 to 3 minutes more. Serve immediately.
