In a bowl, stir the crème fraîche with the shallot, dill and the 2 teaspoons of lemon juice. Season with salt and pepper. Whisk the mixture until thickened, about 30 seconds.
Preheat the oven to 425°. On a box grater, coarsely shred the potatoes. Transfer to a clean kitchen towel and squeeze out as much liquid as possible. In a bowl, combine the potatoes with the 2 tablespoons of melted butter and season with salt and pepper.
In a small nonstick skillet, melt 1 tablespoon of the butter. Add half of the potatoes and pat them into a disk about 1/2 inch thick. Cook over moderately high heat, turning once, until golden, 3 minutes per side. Transfer the potato galette to a baking sheet. Repeat with the remaining 1 tablespoon of butter and potatoes. Bake the galettes for 10 minutes, until cooked through and crisp; transfer to a platter and let stand for 15 minutes.
Spoon the shallot-dill crème onto the galettes and top with the smoked fish. Lightly brush the fish with olive oil and lemon juice and season with salt and pepper. Garnish with chives and caviar. Serve right away.