Baked Potatoes with Broccoli Cheese Sauce

medium Russet potatoes (, scrubbed and rinsed)

butter (, diced into 4 pieces)

all-purpose flour

(scant) onion powder

(scant) garlic powder

milk

Salt and freshly ground black pepper

shredded sharp cheddar ((4 oz))

chopped steamed broccoli florets ((they should be chopped up into very small pieces and then measured)*)