Melt the butter in a 4 to 5 quart pan over medium heat. Add the onions and sweat them, stirring occasionally, 5 minutes.
Raise the heat to medium-high and stir in the flour, mustard, and pepper and cook 2 minutes. Reduce heat to medium, add the milk and stock and cook, stirring frequently, 8 minutes. Add the broccoli and simmer 6 minutes, stirring frequently. Remove from heat, puree 2 cups of soup in a blender, and return pureed soup to pot. Reheat, stirring occasionally, about 2 minutes.