Bring a large pot of water to a boil over high. Add potatoes; return to a boil. Cook until a fork can pierce through about 1/2 inch of the potato with some resistance, about 15 minutes. Transfer potatoes to a cutting board; let cool 20 minutes.
Peel potatoes; discard peels. Grate potatoes on the largest holes of a box grater. Place grated potato in a large bowl; add 1 teaspoon salt, and toss to combine.
Preheat oven to 375°F. Heat 1 tablespoon butter and 1 tablespoon oil in a 9-inch nonstick skillet over medium. Spread grated potato in an even layer in the skillet, and press firmly using a spatula to compact. Cook until bottom is browned and crisp, about 9 minutes, turning skillet occasionally and drizzling 2 tablespoons melted butter around edges of potato round halfway through cooking time.
Slide potato round onto a large plate. Place a second large plate on top, and invert the potato round. Heat remaining 1 tablespoon butter and remaining 1 tablespoon oil in skillet over medium. Slide potato round back into skillet; cook, turning skillet occasionally, until bottom is browned, about 9 minutes.
Place a wire rack on a baking sheet; slide potato round onto a wire rack. Bake in preheated oven until very crisp, about 8 minutes. Sprinkle with remaining 1/4 teaspoon salt. Transfer rösti to a serving platter or cutting board. Top with crème fraîche, leaving a 1-inch border. Top evenly with salmon, red onion, capers, dill, chives, and lemon zest.
