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Preheat the oven to 350 degrees F.
Pierce the potatoes a few times with a fork, then place directly on the center rack in the oven. Bake until tender, about 1 hour.
Meanwhile, melt the butter in a saute pan over medium-high heat. Add the leeks, 1 teaspoon salt and 3 or 4 turns pepper; saute until the leeks are wilted, 3 to 4 minutes. Add the mushrooms and thyme and saute until lightly browned, another 5 to 6 minutes. Remove from the heat and set aside.
Halve the baked potatoes lengthwise and scoop out the flesh, leaving a 1/4-inch-thick shell. Set aside to cool. Reserve the potato innards for another use.
Fill a deep, heavy pot with 3 inches of canola oil; heat to 350 degrees F on a deep-fry thermometer.
Working in batches, fry the potato skins in the oil until crispy and golden brown, 5 to 6 minutes. Drain on paper towels.
Arrange the fried potato skins, scooped-side up, on a baking sheet. Season with 1 teaspoon salt and 3 or 4 turns pepper. Fill the skins with the mushroom-leek filling and top with the Brie and prosciutto. Transfer to the oven and bake until the brie is melted and the prosciutto crispy, 6 to 8 minutes. Garnish with the chives and serve.