In a large pot, melt the butter over medium heat. Add the thinly sliced onions and cook, stirring occasionally, until they are soft and start to caramelize, about 10-15 minutes.

Sprinkle the flour over the caramelized onions and stir well to combine. Cook for 2-3 minutes to cook off the raw flour taste.

Gradually add the chicken and beef broths to the pot, stirring constantly to incorporate the flour and onions into the liquid without clumps.

Bring the mixture to a simmer and let it cook for about 10 minutes, stirring occasionally. The soup should start to thicken slightly.

Reduce the heat to low and stir in the heavy cream and shredded cheddar cheese. Continue cooking, stirring frequently, until the cheese is melted and the soup is heated through. Avoid boiling after adding the cream to prevent curdling.

Taste and adjust seasoning with salt and pepper as needed.

Ladle the soup into bowls and garnish with additional shredded cheddar cheese and chopped chives if desired. Serve hot with a side of crusty bread for dipping.