Heat the oil in a large saucepan over medium heat. Add the carrot, celery, onion, 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until tender, about 8 minutes. Add the garlic and continue to cook for 1 minute.
Add the chicken broth and bring to a boil. Reduce the heat to a simmer and stir in the orzo. Cook until the orzo is just tender, about 7 minutes.
Add the chicken and simmer until heated through. Add the spinach and stir until wilted. Remove from the heat and stir in the dill and lemon zest. Season with additional salt and pepper. Serve sprinkled with additional dill.
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