Heat the oven to 375°F. Line a rimmed baking sheet with a nonstick baking liner or parchment.
Using a mandoline or a sharp knife, cut the potatoes lengthwise into 1/16-inch-thick slices. Soak the 40 nicest, most even slices in a large bowl of cold water until pliable, about 20 minutes.
Drain the potatoes and pat dry with paper towels. Put them in a medium bowl and toss with the oil, 1 tsp. salt, and 1/2 tsp. pepper. Fan 10 potato slices into a 6-inch circle (like a flower) on the prepared sheet. Repeat with the remaining potato slices to make 4 individual circles. Cover the potatoes with parchment and set another baking sheet on top to keep the potatoes flat. Bake until tender, about 15 minutes. Remove the top baking sheet and the parchment and continue to cook until crisp and browned, 15 to 20 minutes more.
Meanwhile, in a small bowl, whisk the crème fraîche, 1-1/2 Tbs. of the chives, the lemon zest, 1/8 tsp. salt, and 1/4 tsp. pepper.
Carefully transfer the potato galettes to individual plates. Place one slice of trout on a potato circle. Add a dollop of crème fraîche, then another slice of trout, another dollop of crème fraîche, and a final slice of trout. Top with a small dollop of crème fraîche and garnish with some of the remaining chives and a grind of pepper. Repeat with the remaining galettes, trout, crème fraîche, chives, and pepper.