In a 5-6 quart Dutch oven over moderate heat, melt the butter in oil. Add the chicken and lightly brown, stirring occasionally, about 5 minutes.
Add the stock, carrots, celery, onion, rice, parsley, salt, and pepper. Cover, bring to a simmer over low heat, then cook until the rice is tender, about 20 minutes.
Blend cornstarch into the half n half with a whisk. Add nutmeg, and whisk again. Stir slowly into the simmering soup. cook, stirring, until the soup returns to a simmer and thickens, about a minute more.