In a large pot, melt the butter over medium heat. Add the onions and cook, stirring occasionally, until soft and starting to caramelize, about 10–15 minutes.
Sprinkle the flour over the onions and stir to combine. Cook 2–3 minutes to remove the raw flour taste.
Gradually add the chicken and beef broth, stirring constantly to blend without clumps.
Bring to a simmer and cook about 10 minutes, stirring occasionally, until slightly thickened.
Reduce heat to low and stir in the heavy cream and cheddar. Cook, stirring frequently, until the cheese melts and the soup is heated through. Do not boil.
Taste and adjust salt and pepper.
Ladle into bowls, garnish with extra cheddar and chopped chives, and serve hot with crusty bread.