Heat butter in a large pot set to medium heat until melted.
Add chopped onion, celery, and carrot and cook until the vegetables have softened, about 5 minutes.
Mix in minced garlic and cook for about 1 minute until fragrant.
Pour the chicken broth into the pot, scraping the bottom to loosen any browned bits.
Add thyme sprigs and increase heat to bring the mixture to a boil.
Lower the heat and let it simmer gently for 20 minutes to build flavor.
Stir in half-and-half and bring the soup back to a gentle simmer.
Remove the thyme sprigs and season with salt and pepper to your liking.
Add shredded chicken and fresh gnocchi to the soup and cook until the gnocchi become tender, following package timing.
Fold in the baby spinach and cook just until it wilts.
Serve the soup hot in bowls and enjoy.