Melt butter in a saucepan over low heat. Remove from heat; let solids sink to bottom. Skim foam. Carefully pour off clarified butter, discarding solids.
Preheat oven to 400 degrees. Slice potatoes a little less than 1/8 inch thick, and lay on a kitchen towel. Heat a large ovenproof skillet (preferably cast-iron) over medium-high heat. Add 2 tablespoons clarified butter; swirl to coat. Arrange some potatoes in concentric circles. Season with 1/4 teaspoon salt and some freshly ground pepper to coat. Repeat with remaining potatoes to form 2 more layers. Pour 3 tablespoons clarified butter down sides of skillet to seep under potatoes. Brush underside of a medium heavy skillet with remaining clarified butter; set over potatoes as a cover. Reduce heat to medium. Cook until underside is golden and crisp, 5 to 7 minutes more. Flip potatoes using 2 metal spatulas. Cover with skillet. Cook until potatoes start to brown, about 4 minutes. Remove top skillet. Bake until underside is crisp, 7 to 9 minutes.
Remove galette from skillet; transfer to a wire rack. Let cool for 10 minutes. Beat cream cheese with creme fraiche until fluffy; spread over potatoes. Top with salmon. Sprinkle with chives. Cut into wedges.