Heat the olive oil until shimmering over medium heat in a large soup pot. Add the onion, carrots, celery, and 1 teaspoon each salt and pepper. Cooking, stirring frequently, until the vegetables are very soft, about 15 minutes.
Add the garlic and cook until fragrant, about 1 minute. Add the stock and bring to a boil.
MAKE AHEAD: Do not add the noodles or parsley. Cool and refrigerate the soup in an airtight container for four days, or in the freezer for up to two months. Reheat on the stove and add the noodles and parsley just before serving.
Add the noodles and cook 6 minutes, then add the chicken and cook about 2 minutes more, until the noodles are cooked through and the chicken is warmed through.
Season to taste with salt and pepper, then stir in the parsley.
Serve topped with Parmesan.
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