The whole family will enjoy this intensely flavorful soup.
In a large, heavy-bottom saucepan, melt the butter over medium heat. Add flour and chopped onions; saute, stirring occasionally, until onions are soft and translucent, about 15 minutes.
Pour chicken and beef stocks into the saucepan, and stir to combine. Bring mixture to a boil. Reduce heat to low, and simmer 15 minutes. Add salt and pepper.
Heat broiler. Ladle soup into six 16-ounce ovenproof bowls, and float 3 slices of bread on top of each. Sprinkle the cheese over bread slices in each bowl, dividing evenly. Place bowls on a rimmed baking sheet, and broil until cheese is melted and bubbling, about 2 minutes. Let soup cool slightly before serving.
If you don't have oven-proof bowls, you can toast the cheese-topped bread slices on a baking sheet under the broiler before adding to the soup.