Arrange the racks in your oven so that one is in the middle and one is underneath in the bottom half. Preheat the oven to 400degrees and line a sheet pan with aluminum foil. Use a fork to poke holes around the surface of your potatoes, about 6-8 times each. Place the sheet pan on the bottom rack and position the potatoes directly on the rack above it (this way if there are any drippings they will fall on the lined sheet pan and not your oven). Bake for an hour or until cooked through. Remove the potatoes from the oven and set aside to cool. Remove the lined sheet pan from the oven as well for later use. Keep the oven on at 400 degrees.
Place the broccoli florets in a microwave-safe bowl and cover the bowl with a damp paper towel. Microwave for 2 minutes on high.
When the potatoes are cooled enough to easily handle (about 10 minutes), slice them in half lengthwise. Use a spoon to scoop out the insides of each potato half into a mixing bowl, leaving a ¼” border around the outside of each potato shell. Line the potato shells up on the reserved sheet pan from step one.
Add the light butter and sour cream to the potatoes in the mixing bowl and sprinkle in the salt and pepper. Use a potato masher or a fork to mash together. Add *most* of the shredded cheddar cheese (I reserve a bit to sprinkle over the top) and stir the cheese into the mashed potatoes. Add the cooked broccoli florets from step two and stir to combine.
Spoon the mashed potato filling evenly into the potato shells and sprinkle the remaining cheese over the top. Place the sheet pan back into the 400 degree oven and bake for about 10 minutes until the cheese is melted and the tops are golden. Serve warm.