Place the first seven ingredients in a stockpot. Bring to a boil. Reduce heat; cover and simmer for 10-15 minutes or until vegetables are tender. Add beef and rice.
In a small bowl, combine flour and milk until smooth. Stir into the soup. Bring to a boil; cook and stir for 2 minutes or until thickened. Stir in the butter and seasonings.
Combine the baking mix, cheese and milk. Drop by tablespoonfuls onto simmering soup. Cover and simmer for 12 minutes or until a toothpick inserted in a dumpling comes out clean (do not lift the cover while simmering).