Melt butter in a large saucepan or soup pot.
Add onion and carrot; cook over medium-low heat for 7 to 8 minutes, or until vegetables are tender.
Stir in flour and chili powder.
Slowly add broth and continue to stir until everything is well combined.
Add apple to soup and cover.
Reduce heat to a simmer and continue to cook for about 10 minutes, or until soup is thick and apples are tender. Stir occasionally.
Slowly stir in apple juice and shredded cheddar cheese.
Continue to cook and stir until cheese is completely melted.
Remove from heat.
Let cool for 2 minutes.
Transfer to a blender and blend until smooth.
Taste for salt and pepper.
Ladle soup into bowls and garnish with diced apple.