Preheat an oven to 425ºF.
Scrub the potatoes of any excess dirt and prick them each several times with a fork. Place them on a baking sheet, drizzle with EVOO and season with salt and freshly ground black pepper, tossing lightly to coat.
Bake in the oven until tender, about 45-60 minutes depending on the size of the potato. Remove the baked potatoes from the oven and let cool enough to handle.
Once the potatoes are cool, prepare the broccoli by placing high-sided pan with a lid over high heat with about an inch of water (if your pan doesn’t have a lid, just cover it with foil). Cook the broccoli in the pan until tender, 3-5 minutes. Remove the broccoli from the pan with a slotted spoon to a medium-size mixing bowl.