Heat the butter in a large stock pot over medium-low heat. Add in the onion, carrots, celery and garlic with a big pinch of salt and pepper - I’d say at least ½ teaspoon salt and pepper to start. Cook, stirring occasionally, until the vegetables soften. Stir in the garlic cloves and poultry seasoning. Stir in the shredded chicken.
Add in the chicken stock and the parmesan rind. Bring the mixture to a boil, then reduce it to a simmer. Stir in the orzo pasta. Cook uncovered, stirring occasionally, until the orzo is al dente, about 8 minutes.
In a shaker bottle even a bowl, shake (or whisk) together the milk and flour to form a slurry. Pour this into the soup and simmer for another 1 to 2 minutes. Taste and season with more salt and pepper as needed. If the parmesan rind didn’t melt entirely, you can scoop it out at this point. Stir in the fresh parsley.
Serve immediately with a lemon wedge for sprinkling, parmesan cheese for topping and parsley sprinkled on top.
Note: if you want to prep this for meals ahead of time, or want to have it for leftovers, I suggest cooking the orzo separately as it will soak up most of the broth as it sits overnight. Combine the soup and orzo and reheat to enjoy!