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In a large bowl, whisk the flour, 1 ¼ tsp salt and 1 ¼ tsp pepper. Add the chicken and toss to coat. Remove from flour and shake off excess. Discard flour.
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In a large soup pot, heat butter over medium-high heat until melted. Add the onion, celery and bay leaf. Sauté until tender, about 5 minutes.
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Add the mushrooms and sauté until they give up most of their liquid and are lightly browned, about 5 minutes. Next add the garlic, sauté until lightly browned, about 30 seconds.
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Add the chicken and cook, stirring occasionally, until lightly browned, about 3 minutes. (The chicken will not be cooked through.)
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Add wine (if using) and cook until almost entirely evaporated, about 1 minute. Stir in broth and bring to a simmer. Reduce the heat and simmer 10 minutes.
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Stir in cooked rice, milk and cream. Simmer, stirring occasionally, until the flavors come together, about 20 minutes. Season with ½ tsp salt and ½ tsp pepper. Remove and discard bay leaf. Serve hot with parsley.