Prick the potatoes all over with a fork, then place them directly on the middle oven rack and bake until soft when squeezed lightly and the skin is crisp, about 1 hour. Remove and let cool to handle, 15 minutes. (Leave the oven on.)
Meanwhile, heat a large, dry skillet over medium-high heat. Lay the chicken skin flat on a work surface and sprinkle with salt and pepper on both sides. Place the skin flesh-side down in the skillet and fry like bacon, placing a heatproof plate or bowl on top for even cooking and extra crispness, until golden brown, about 5 minutes per side. Remove to paper towels reserving the chicken fat (schmaltz). Break the skin into pieces when cool enough to handle.
Slice the potatoes in half. Scoop out the middles (save for another use such as pancakes, soup or mashed potatoes). Lay cut-side up on a baking sheet. Brush with the schmaltz and sprinkle with more salt and pepper. Bake until lightly brown, about 10 minutes.
Turn the oven to broil. Combine both cheeses, the olives and scallion whites in a small bowl. Divide into the scooped potatoes and broil until the cheese is bubbly, 2 to 3 minutes. Garnish with the chicken skin, scallion greens, sour cream and a dash of sriracha.