Wash your hands with soap and water.
Rinse potatoes well under running water and scrub with a clean vegetable brush to remove any dirt; try not to break the skin. Remove any bruises or discolored spots with the tip of a knife. Pat dry with a clean paper towel.
Microwave potatoes according to the manufacturers' directions.
Heat 3/4 cup of the milk in a small saucepan over medium heat until steaming, about 4 minutes.
Stir together flour and the remaining 1/4 cup milk in a cup until smooth. Add to the hot milk, stirring constantly until the mixture boils and thickens. Remove from heat and stir in cheese. Season with salt and cayenne.
Distribute broccoli over the baked potatoes, top with the cheese sauce and serve.