Scrub and dry the potatoes. Preheat the oven to 425 degrees. Place the potatoes on a small, foil-lined baking sheet, pierce each several times with a fork. Bake until fork-tender, 1 hour.
Drop the broccoli into a small saucepan of boiling water (ask a grown-up to help with this). Cook until crisp-tender, 3 to 4 minutes. Drain in a strainer, then rinse with cold water.
Cut the warm potatoes in half lengthwise. Scoop out most of the flesh, leaving about 1/4 inch of potato in the skins, and transfer to a bowl. Return the scooped-out skins to the baking sheet.
Smash the potatoes with a potato masher until fairly smooth; season with 1/2 tsp. salt and 1/4 tsp. pepper. Stir in the sour cream and 1/2 tsp. tbsp. butter.
Stir the cooked, drained broccoli into the potato mixture. Ask someone (like your twin sister!) to hold the bowl.
Melt 1/2 tbsp. butter in a medium skillet over medium-high heat. Add the ham and cook until golden, 4 minutes. Spoon the ham into the potato shells.
Add 1 tbsp. butter to the skillet; melt over medium heat. Dump in the beans; mash. Add 1/3 cup water and mash until the beans are thick; season.
Spoon the beans on top of the ham in the potato shells. Sprinkle 1/4 cup cheese over each shell.
Mound the potato-broccoli mixture on top and sprinkle with the remaining cheese. Bake until the cheese is golden, 20 to 25 minutes.