Preheat the grill to medium-high heat (about 450 degrees F).
Slice the potatoes in half lengthwise. Brush all over with oil and place on the grill skin side down. Close the lid and cook for 30 minutes.
Remove the potatoes from the grill, leaving the grill on, and let the potatoes cool slightly. Scoop out the insides, leaving the skins intact with a ¼-inch-thickness of potato flesh.
Place the scooped-out potato flesh in a large bowl and mash with a fork. Add the scallions (reserving some for garnish), mushrooms, chives, cheddar, sour cream, and salt and pepper to taste, and stir to combine.
Divide the stuffing equally among the potato shells. Place the stuffed potatoes on the grill, close the lid, and cook until the cheese has melted about 10 minutes. Serve, garnished with scallions and chives.
