In a large saucepan, melt 3 tablespoons of butter over medium heat. Add the thinly sliced onions and cook, stirring frequently, until they become soft and translucent but not browned. This should take about 10-15 minutes.

Once the onions are tender, pour in the chicken broth. Add the chicken bouillon cubes, freshly ground black pepper, and ¼ teaspoon of salt. Stir well to combine and let the mixture heat through.

In a medium saucepan, melt the remaining 3 tablespoons of butter over medium heat. Stir in the all-purpose flour, cooking until the mixture thickens and pulls away from the sides of the pan. Gradually add the whole milk, a little at a time, stirring constantly to prevent lumps. Add the remaining ¼ teaspoon of salt. Continue stirring until the sauce reaches a thick, pudding-like consistency. Remove from heat and set aside.

Transfer the white sauce into the saucepan with the onion mixture. Add the cubed Velveeta cheese. Simmer over medium-low heat, stirring constantly, until the cheese has melted and all ingredients are well combined. Reduce the heat to low and let the soup cook for an additional 30 to 45 minutes, stirring occasionally.

Ladle the hot soup into bowls. Garnish with shredded cheddar cheese and serve with slices of dark Russian bread, if desired.