Fill a large saucepan with cold water and add the potatoes. Bring to a boil over medium-high heat, and then lower to a simmer and cook until a fork easily pierces through the potatoes, about 30 minutes. Drain in a colander and cool slightly.
While the potatoes are cooling, make the orange-vanilla sugar. Put the orange zest on a microwave-safe paper-towel-lined plate and microwave in 30-second intervals until the zest is dry and almost brittle, 3 to 4 minutes total. Transfer to a food processor with the sugar and vanilla bean and grind until the sugar is slightly powdery with bits of vanilla bean and orange zest throughout. Pour into a large mixing bowl.
When the potatoes are cool enough to handle, peel and cut into 1-inch chunks. Run the potatoes through a ricer into a large mixing bowl, and then stir in the flour, salt and egg. Knead well to form a smooth dough.
Turn out the dough onto a lightly-floured surface and roll into a 10-by-8-inch 1/2-inch-thick rectangle. Cut the dough in half lengthwise using a bench scraper, and then cut across the dough widthwise into 1/2-inch sections to produce 34 rectangles. Roll the rectangles into 6-inch-long dumplings with tapered ends using your hands. Transfer to a parchment-lined flour-dusted baking sheet and let the dumplings rest at room temperature for 15 minutes.
Meanwhile, fill a large Dutch oven with 2 inches of oil and bring to 375 degrees F over medium-high heat. Fry the dumplings in small batches until very golden brown and crispy, 4 to 5 minutes, and then transfer to a paper-towel-lined baking sheet to drain. Toss the hot dumplings in the orange-vanilla sugar to coat and serve immediately.
Cook's Note: Keep the leftover orange-vanilla sugar in an airtight container and use it in place of regular sugar in other desserts.