Melt the butter in a large pot over medium heat. Add the onion and cook for a few minutes, stirring occasionally, until it begins to soften.

Sprinkle the flour over the onions and stir continuously until the flour is fully mixed with the butter and the mixture begins to bubble.

Slowly pour in the milk while stirring to keep the mixture smooth. Increase the heat to high and bring it to a boil.

Once boiling, reduce the heat to low and let the soup simmer for a couple of minutes, stirring occasionally as it begins to thicken.

Add the Velveeta, broccoli, and pepper. Cook over low heat, stirring often, until the cheese is completely melted. As the broccoli heats through, use your spoon to break any larger pieces into smaller, bite-size pieces.

Once the cheese has melted and the soup is smooth and creamy, simmer for another 5 minutes. Stir occasionally, then serve warm.