Heat olive oil in a large pot over medium heat. Add onion, carrot, and celery and sauté for 5-7 minutes until softened. Add garlic and cook for 1 minute more.
Push vegetables to the side and melt butter in the empty space. Sprinkle flour over butter and whisk continuously for 2 minutes until golden.
Slowly pour in chicken broth while whisking to avoid lumps. Bring to a simmer.
Stir in heavy cream, milk, and buffalo sauce. Adjust buffalo sauce to taste.
Add shredded chicken, smoked paprika, garlic powder, onion powder, and cayenne pepper. Season with salt and pepper.
Reduce heat to low and gradually add cheddar cheese and cream cheese, stirring until melted and smooth.
Taste and adjust seasoning as needed. Garnish with green onions and serve immediately.