Preheat oven to 350°F.
Place potatoes on a baking sheet. Roast until fork-tender, 1 hour to 1 hour, 15 minutes. Let cool until room temperature, about 1 hour. (Do not turn oven off.) Peel cooled potatoes and grate the potatoes on large holes of a box grater. Transfer grated potatoes to a large heatproof bowl; set aside.
Melt butter in a small saucepan over medium. Add shallot; cook, stirring often, until softened, 5 to 6 minutes. Pour shallot-butter mixture into bowl with potatoes. Add chives, potato starch, salt, and pepper; stir mixture until well combined.
Line a 9- x 13-inch baking pan with parchment paper, and grease with oil. Spread potato mixture evenly on baking sheet; using your hands or the bottom of a flat glass, press mixture firmly to compact. Cover mixture with another sheet of parchment paper, and top with a separate 9- x 13-inch baking sheet. Transfer to oven, and bake at 350°F for 30 minutes. Transfer to a wire rack (with top baking sheet still in place), and let cool 1 hour. Transfer to refrigerator; place a large can or weight on the top baking sheet to compress potato mixture. Chill at least 8 hours or up to 24 hours.
Add oil to a large Dutch oven to a depth of 2 inches; heat over medium until oil reaches 360°F. Meanwhile, uncover potato mixture, and cut evenly into 26 (1- x 4 1/3-inch) sticks.
Working in batches of about 6 sticks, fry potato sticks, turning occasionally, until crisp and golden brown, about 1 minute, 30 seconds per batch. Using a spider, transfer hashbrowns to a paper towel-lined baking sheet, and season to taste with salt and pepper.
