Combine chicken broth, butternut squash, and sweet potatoes in a Dutch oven; bring to a boil. Cover, reduce heat, and simmer 15 minutes or until squash and potato are tender. Remove squash and potato from pan with a slotted spoon, reserving broth. Mash squash mixture to desired consistency.
While squash and potato cook, melt 1 teaspoon butter in a medium nonstick skillet over medium-low heat. Add the onion and apple; cook 20 minutes or until lightly browned, stirring occasionally. Remove onion mixture from pan; set aside. Increase heat to medium-high. Melt remaining 3 teaspoons butter in pan. Add flour; cook for 5 minutes or until golden brown, stirring constantly with a whisk. Gradually add 1 cup reserved broth, and cook for 3 minutes or until slightly thickened.
Add the thickened broth mixture, mashed squash mixture, onion mixture, salt, and pepper to remaining broth in Dutch oven. Cook over medium heat 5 minutes, stirring occasionally.