Position rack in center of oven. Preheat oven to 400°F/200°C. Place potatoes on rack and bake for 45 minutes to 1 hour or until tender when pierced with a sharp knife. Alternatively, microwave until cooked through. Set aside until cool enough to handle.

While the potatoes are cooking, bring a small pot of water to a boil and blanch the broccoli for about 1 minute, or until crisp yet tender and still bright green. Drain and run under cold water to stop cooking. Blot dry with paper towels and set aside.

Meanwhile, stir together the chicken, half of each of the cheeses and scallions, and all of the sour cream and oil in a mixing bowl until combined; set aside.

When the potatoes are cool enough to handle, slice the top off of each one and scoop out the flesh leaving a 1/4 inch (6 mm) wall all around, adding the potato flesh to the bowl of chicken. Take care not to tear through the potato skins. Mix the potato into the chicken mixture, season liberally with salt and pepper to taste, then gently fold in broccoli. Stuff this mixture back into the potato jackets, mounding high. Place potatoes in a roasting pan. Bake in 400°F/200°C oven until heated through, about 10 to 15 minutes. Sprinkle remaining cheeses on top for the last 5 minutes of baking; the cheeses should be nicely melted. This is a dish that offers some leeway in baking time. Just make sure innards are hot and cheese is melty! (Yes, in a pinch the re-heating can be done in the microwave; just make sure your pan is microwave safe). Sprinkle with reserved scallions and serve immediately.