Preheat oven to 350°F. Place potatoes on a baking sheet; roast until tender when pierced with a fork, about 1 hour and 15 minutes. Let cool 30 minutes. (Leave oven on.) Peel potatoes; discard skins. Grate potatoes on large holes of a box grater into 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 mixture into bowl with potatoes. Add chives, potato starch, salt, and pepper; stir mixture until well combined.

Line a 13- x 9-inch baking sheet with parchment paper, and grease with oil. Spread potato mixture evenly on baking sheet; press firmly to compact. Cover mixture with another sheet of parchment paper, and top with a separate 13- x 9-inch baking sheet.

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 top baking sheet to compress potato mixture. Chill at least 8 hours or up to 24 hours. Cut chilled potato mixture into 24 (about 1- x 4-inch) sticks.

Heat 2 inches of oil in a large, deep pot over medium to 360°F. Working in batches fry potato sticks, turning occasionally, unt crisp and golden brown, about 4 minutes. Transfer hash browns to a paper towel-lined baking sheet, and season with salt and pepper to taste.

Serve hash browns topped with crème fraîche and trout roe. Garnish with chives. Serve hot.

Note: Hash browns can be prepared through step 4 and frozen on a parchment paper-lined baking sheet until hard, about 30 minutes. Transfer to a ziplock plastic freezer bag, and freeze up to 1 month. Fry directly from frozen. increasing frying time to about 5 minutes per batch.