Broccoli roasts on a lower rack while the potatoes are baking here. The charred florets are then teamed with a rich cheese sauce atop the fluffy spuds.
Preheat oven to 400 degrees. Thoroughly wash and dry potatoes. Prick each in a few spots with a fork. Brush with oil and season with salt.
Place potatoes directly on rack and bake until soft and easily pierced with the tip of a knife, 1 hour to 1 hour, 15 minutes, depending on size. Meanwhile, toss broccoli with oil. Spread onto a rimmed baking sheet; season with salt and pepper. When potatoes are halfway through baking, place sheet on rack in lower third. Cook until broccoli is tender and brown in places, about 20 minutes. Remove from oven and set aside.
Melt butter in a saucepan over medium-high heat. Whisk in flour. Cook until golden, about 1 minute. Whisk in milk. Boil, whisking, until thickened slightly, 3 to 4 minutes. Remove from heat. Stir in cheddar, cayenne, and 1/2 teaspoon salt until smooth.
Remove potatoes from oven. Grab a hot potato with a clean kitchen towel and bang it down firmly on the counter once to fluff up the flesh. Repeat with remaining potatoes. Cut open and season generously with salt and pepper. Top with broccoli and drizzle with cheese sauce; serve.