Finely grate the Parmesan cheese, reserving the rind. Pour the rice into a fine-mesh sieve and rinse under cold water until the water runs clear. Set the sieve holding the rice over a bowl, so that it continues to drain while you prepare the other ingredients.
Melt the butter in a heavy stockpot over a medium-high heat. When it starts to foam, add the rice to the pot and sauté in the hot fat until the tips of the grains become translucent, about 2 minutes. Add the sliced leek and chopped celery to the pot, stirring continuously until the vegetables soften, about 3 minutes. Stir in the cooked chicken and cook for a further 2 minutes.
Pour the broth into the stockpot, then drop in the Parmesan rind. Bring to a simmer, then reduce the heat to medium. If you’re using fresh peas, add them to the pot after the rice has been cooking for 10 minutes. Continue to simmer the soup, covered, until the rice and fresh peas are tender, about 10 minutes. If you’re using frozen peas, simply cook the rice until tender – about 15–20 minutes – then stir the frozen peas into the soup. Simmer for a further 5 minutes, until the peas are warmed through.
Turn off the heat. Using a pair of tongs, remove the Parmesan rind and discard. Pour the lemon juice through a fine-mesh strainer to remove any pips and add to the soup. Sprinkle in the parsley and lemon zest. Season to taste with salt and pepper. Ladle the soup into bowls and top with grated Parmesan cheese just before serving.
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