Warm the olive oil in a large soup pot over medium heat, then cook the onion for 5 minutes before adding the carrots and celery for 3 minutes. The onion edges should look translucent and the vegetables should smell sweet and fragrant.
Stir in the butter and flour and cook for 1 minute, then add the chicken, garlic, Italian seasoning, paprika, thyme, chili flakes, salt, pepper, and 4 cups broth. The flour mixture should turn light golden and coat the ingredients before the broth comes to a gentle simmer.
Simmer over medium heat for 20 minutes, then lift out the cooked chicken and shred it with two forks. The chicken should pull apart easily without resistance when it is ready.
Return the shredded chicken to the pot and stir in the spinach, sun-dried tomatoes, coconut milk, cream, or milk, parmesan, and gnocchi. Cook for about 5 minutes, until the spinach wilts, the cheese melts, and the gnocchi floats and feels pillowy.
Add more broth if the soup becomes thicker than you like, then ladle it into bowls and top with parmesan. The finished broth should look creamy and fluid enough to surround the vegetables and dumplings.