Heat olive oil and butter in a large Dutch oven or pot over medium heat.
Add the diced onion, celery, and shredded carrots. Sauté for 5 minutes until the vegetables begin to soften.
Stir in the minced garlic and cook for 1 minute until fragrant.
Whisk in the all-purpose flour and cook for 2 minutes, stirring constantly to form a light roux.
Gradually pour in the chicken broth and half and half, whisking continuously to ensure no lumps form.
Add the dried thyme, salt, and pepper. Bring the mixture to a gentle simmer.
Add the potato gnocchi to the pot and cook for 4 to 5 minutes, or until the gnocchi float to the surface and are tender.
Stir in the shredded chicken and the chopped spinach.
Continue to simmer for 2 to 3 minutes until the spinach is wilted and the soup reaches the desired thickness.
Taste and adjust seasoning with additional salt and pepper if necessary.
Serve hot, optionally garnished with grated Parmesan cheese.