Melt the butter in a Dutch oven over medium heat, add the onion, celery and carrot, and cook for 6 minutes until the onion looks translucent.
Sprinkle in the flour and stir for 1 minute until no dry pockets remain.
Whisk in the broth gradually, bring it to a gentle simmer and cook for 5 minutes until lightly thickened.
Add the chicken and gnocchi, then simmer for 5 to 7 minutes until the gnocchi are tender and the chicken reaches 165°F.
Lower the heat, stir in the half-and-half, spinach, salt and pepper, and cook for 2 minutes without boiling until the spinach wilts.