Position one oven rack in the middle and a second rack about 8 inches from the broiler element. Heat the oven to 400°F. Rub the potatoes with vegetable oil and roast on a foil-lined baking sheet until a skewer can be inserted with little resistance, 60 to 70 minutes. Let the potatoes cool for 10 minutes.

Melt half of the butter in a large skillet over medium-high heat. Add the broccoli and a portion of the salt and cook until lightly browned, about 2 minutes. Add the water, cover, and cook until the broccoli is crisp-tender, then uncover and let the water evaporate completely. Transfer the broccoli to a bowl and drizzle with lemon juice.

Halve each potato lengthwise. Using a spoon, scoop the flesh from each half into a bowl, leaving about 3/8-inch thickness of flesh attached to the skin. Place the shells, cut-sides up, back on the baking sheet and return them to the oven until the interiors are dry and the edges are slightly crisp, about 10 minutes.

Add the remaining butter to the bowl of potato flesh and mash with a fork or potato masher until smooth. Stir in the larger portion of cheddar, the scallions, sour cream, half-and-half, mustard, and remaining salt. Season with pepper to taste, then gently fold in the cooked broccoli.

Remove the shells from the oven and heat the broiler. Mound the filling into the shells and sprinkle with the remaining cheddar. Broil until the cheese is melted and spotty brown, 6 to 10 minutes. Let the potatoes rest for 5 minutes before serving so the filling sets.