In 2-quart saucepan, place butter and sliced onions. Cook at low to medium heat, stirring frequently, until soft and clear but not brown. Add chicken broth from can, chicken bouillon cubes, salt, and pepper and stir until completely heated through.
In a 1-quart saucepan, melt butter and add flour. Cook on medium heat until the flour turns thick and comes away from the side of the saucepan. Pour milk into flour a little at a time and stir constantly; add salt. The mixture should thicken and become like thick pudding. Remember to stir constantly, taking care not to let mixture lump. Remove from heat and set aside until ready to use for the soup.
Finishing the soup
Add white sauce and Velveeta cheese. White sauce will be thick because it has been removed from the heat. Simmer on medium-low heat until the cheese is melted and all ingredients are blended, stirring constantly. Turn temperature to warm and let cook for additional 30-45 minutes. Serve with a garnish of shredded cheddar cheese and a couple of slices of warm, dark Russian bread.