In a large pot over medium heat, melt the butter. Add onions, celery, and carrots, and cook until they soften, which takes about 5 minutes. Toss in the minced garlic and cook until it smells amazing, around 1 minute.
Pour in the chicken broth and deglaze the pot, scraping up any tasty bits stuck to the bottom with a wooden spoon. Toss in the thyme sprigs and bring everything to a boil.
Reduce the heat to low and let the soup simmer for 20 minutes, allowing all those wonderful flavors to mingle. Stir in the half-and-half and let the soup gently simmer again. Fish out the thyme sprigs and season the soup with salt and pepper to taste.
Add the shredded chicken and gnocchi to the pot and let them simmer until the gnocchi are tender, following the package directions. Toss in the spinach and cook until it just wilts. Ladle the soup into bowls and serve up a comforting bowl of goodness!