Adjust an oven rack to the middle position and heat the oven to 400°F. Rub the potatoes with oil and prick them several times with a fork. Place the potatoes on a wire rack set in a rimmed baking sheet and bake until the skins are crisp and a paring knife glides easily through the flesh, 50 to 60 minutes.

Melt a portion of the butter in a large skillet over medium-high heat. Add the broccoli, salt, pepper, and water; cover and cook until the broccoli is bright green and just tender, about 3 to 5 minutes. Remove the lid and continue to cook until the water has evaporated. Stir in the lemon juice, then transfer the broccoli to a bowl and set aside.

Let the potatoes cool for 10 minutes so they are easier to handle. Cut each potato in half lengthwise. Scoop the flesh into a medium bowl, leaving a 1/4-inch-thick shell. Return the empty shells to the wire rack and bake until they are dry and slightly crisp, about 10 minutes.

Mash the potato flesh with the remaining butter until smooth. Stir in the majority of the cheese, the scallions, sour cream, half-and-half, dry mustard, salt, and pepper. Gently fold in the cooked broccoli until evenly distributed.

Adjust the oven rack to the upper-middle position and heat the broiler. Divide the potato mixture evenly among the shells, mounding it slightly. Sprinkle the remaining cheese over the top. Broil until the cheese is melted and bubbly with some browned spots, about 3 minutes. Serve hot.