Peel the potatoes, then cut them lengthwise in long matchsticks, using the finest julienne blade of a mandoline. Spread the potatoes out neatly in one layer on a clean kitchen towel or paper towels, roll the towel up, and squeeze lightly to dry the potatoes without breaking them up. Put the potatoes into a bowl and toss them with 1 teaspoon salt and 1/2 teaspoon pepper.
Heat 2 tablespoons oil in a small (8-inch) omelet pan (a saute pan with sloped sides) over medium heat. Add half of the potatoes to the pan and press lightly with a large metal spatula to make the shreds lie flat. Cook them undisturbed for 5 minutes. Drizzle 1 tablespoon of the butter around the edge of the pan and cook the potatoes for another 3 to 5 minutes, until nicely browned on the bottom. Loosen the galette gently around the edge with the metal spatula and flip it over. (Be careful! The oil and butter underneath are very hot!) Add another tablespoon of butter around the edge and cook the galette for 4 to 5 minutes longer, until the second side is nicely browned. With the metal spatula, transfer the galette to a paper towel-lined plate and set aside. Repeat with the rest of the potatoes and butter to make a second galette.
Place the two galettes on a cutting board. Spread each galette with 2 tablespoons of the creme fraiche and a layer of smoked salmon, covering the galette completely. Sprinkle with minced chives and sprinkle lightly with salt. Use a large chef’s knife to cut each galette into 6 wedges and serve hot.
To make ahead: Prepare the potato galettes up to an hour ahead and reheat directly on an oven rack in a 400-degree F oven for 5 to 10 minutes, until heated through. Assemble just before serving.