Preheat your oven to 375 degrees Fahrenheit. With your hands, cover the potatoes in a thin layer of the vegetable oil. Place on a baking pan and cook for 1 hour.
While the potatoes cook, place the broccoli in a small saucepan. Fill the pan with water halfway up to the broccoli and cook over medium heat, until the water reaches a simmer. Lower the heat slightly and cover, cooking for 3-4 minutes, or until the broccoli reaches your desired tenderness. Drain and set aside.
Cut the cooked potatoes in half. With a spoon, scoop out the insides (carefully if they are still hot and steamy), leaving a thin layer of potato left on the skin intact. Place the potato that you scooped out into a large bowl or bowl of a stand mixer. Add the broccoli, 3/4 cup of cheddar, and spices. Mash with a potato masher or with the paddle of a stand mixer on low, until well combined.
Scoop the mixture back into the cored potatoes, filling them as much as possible. Sprinkle with the leftover 1/4 cup of cheese. Bake in the oven at the same heat for 15 minutes. Serve warm right out of the oven. They can also be stored wrapped tightly in the refrigerator and heated in the microwave.