In 4-quart Dutch oven, melt butter over medium-low heat. Add apples and onion. Cook 12 to 15 minutes, stirring frequently, until soft but not brown. Add carrot and cook 5 minutes longer or until carrot just begins to soften. Increase heat to medium. Add flour and cook 1 minute, stirring constantly as mixture thickens. Add 1/2 cup of the broth, stirring constantly with wire whisk to prevent lumps. As soon as broth is incorporated (about 20 seconds), add another 1/2 cup broth and stir again until incorporated. Add remaining broth and the apple juice.