Melt butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add 1 cup milk; cook over medium heat, stirring constantly, until thickened and bubbly. Gradually stir 2 tablespoons hot mixture into 1 beaten egg; add to remaining hot mixture, stirring constantly. Stir in paprika. Add ham and rice; mix well. Set aside to cool.
Shape mixture into 12 (2 1/2- inch) croquettes. Combine 1 beaten egg and 1 tablespoon milk; mix well. Dip each croquette into egg mixture, and coat with breadcrumbs. Fry croquettes in hot oil until brown on both sides, turning once.