In a large pot, melt the butter and add the onion, carrots and celery. Cook, stirring often, until the vegetables are mostly tender, 5-7 minutes.
Add the flour and stir to combine, cooking another 30 seconds or so.
Gradually add the chicken broth, stirring or whisking constantly to avoid lumps. Stir in the salt, pepper, and cooked rice.
Bring the soup to a gentle simmer, and cook for 2-3 minutes, stirring often to make sure it's not sticking on the bottom.
Add the chicken. Stir in the half-and-half or milk. Simmer for 3-4 more minutes. Add additional salt and pepper to taste.
Serve, garnishing with fresh chives (optional).
