In a 2-quart saucepan, melt 2 tbsp butter over medium heat.
Add the sliced onions and cook, stirring frequently, until soft and translucent, but not browned.
Add chicken broth, bouillon cubes, salt, and pepper to the saucepan with the onions. Stir and heat until warmed through.
For the thick white sauce, melt 3 tbsp butter in a separate saucepan over medium heat.
Stir in flour, cooking until it forms a smooth, thick mixture that pulls from the pan’s sides.
Slowly pour in the whole milk, stirring continuously to avoid lumps.
Add 1/4 tsp salt and cook until the sauce thickens to a pudding-like consistency. Remove from heat.
Combine the Thick White Sauce and diced Velveeta cheese with the onion mixture.
Stir over medium-low heat until the cheese melts and blends smoothly.
Reduce the heat to low and let the soup simmer for 30-45 minutes, stirring occasionally to allow flavors to meld beautifully.
Serve warm with shredded cheddar and dark Russian bread.