Preheat oven to 400 degrees. Bake potatoes until tender, about 1 hour. Let cool slightly on a wire rack.
Prepare an ice-water bath; set aside. Bring a large pot of water to a boil. Add 1 tablespoon salt. Add broccoli, and cook until very tender, about 8 minutes. Transfer to ice-water bath to stop cooking. Drain well. Set aside.
Cut potatoes lengthwise, removing 1/2 inch off tops; set tops aside. Holding potatoes with a kitchen towel to protect your hand, scoop out flesh, leaving about a 1/2-inch-thick shell on skins; discard flesh of 2 potatoes. Pass remaining flesh through a ricer into a large bowl.
Add broccoli to bowl with potatoes. Using a pastry blender or potato masher, mash broccoli and potatoes together. Add butter, oil, 2 teaspoons salt, and 1/2 teaspoon pepper; mash until combined. Taste and adjust for salt and pepper. Season potato shells with salt and pepper. Spoon potato mixture into shells. Cover with reserved tops. Transfer to a rimmed baking sheet. Cover sheet with foil, and bake until heated through, about 15 minutes. Serve immediately.