Place 3 tablespoons of butter and the onions in a 2-quart saucepan and cook over medium-low heat, stirring frequently, until the onions are soft and translucent but have not browned. Add the chicken broth, bouillon cubes, pepper, and 1/4 teaspoon of salt and stir until heated through.
Melt 3 tablespoons of butter in a 1-quart saucepan over medium heat. Add the flour and cook until it thickens and comes away from the sides of the saucepan. Add the milk a little at a time, stirring constantly. Then add 1/4 teaspoon of salt. Keep stirring to prevent any lumps from forming as the mixture thickens to the consistency of thick pudding. Remove from the heat and set aside until ready to use.
Transfer the contents of the smaller saucepan to the larger saucepan along with the Velveeta cheese. Simmer on medium-low heat, stirring constantly, until the cheese melts and all the ingredients are blended. Turn the temperature to warm and cook for an additional 30 to 45 minutes.
Serve with a garnish of shredded cheddar cheese and a few slices of warm, dark Russian bread.