In a 10-inch nonstick skillet, melt the butter. Add the leeks and cook over moderate heat until softened, about 7 minutes. Scrape the leeks into a large bowl and toss them with the shredded potatoes and 1 teaspoon of salt. Wipe the skillet out with a paper towel.
Heat the olive oil in the skillet. Add the potato-leek mixture, pressing it into a cake with a spatula. Cook the rösti over moderate heat, pressing down on it occasionally with the spatula, until the bottom is nicely browned, 10 to 12 minutes; reduce the heat to low if the rösti seems to be browning too quickly. Slide the rösti out onto a plate, then carefully flip it back into the pan with the browned side up. Press down with the spatula and cook it over moderate heat, lowering the heat if necessary, until tender throughout and well browned on the bottom, about 12 minutes longer.
Slide the potato-leek rösti onto a cutting board and cut it into strips or wedges. Season with salt and serve immediately.
The rösti can be kept at room temperature for up to 2 hours. Recrisp on a baking sheet in a 400° oven.
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