Add the butter to a large pot and melt it over medium heat.

Add the diced onion and cook for a few minutes, until softened.

Sprinkle in the flour and stir constantly. Cook until the mixture is bubbly, about a minute.

Pour in the milk and bring it to a boil over high heat, then reduce to low and simmer for a couple of minutes.

Add the cubed Velveeta, thawed broccoli, and pepper. Stir often as the cheese melts, breaking apart any large pieces of broccoli as it cooks.

Once the cheese is fully melted, let it simmer for 5 more minutes before serving.