Don't peel the potatoes, just trim out the eyes. Put them in a pot and cover with water. Salt the water, bring it to a boil, and simmer the potatoes for 15 minutes. Scoop them out of the water and let them cool.
When the potatoes are cool, grate them. I did this by hand using the coarsest teeth on the grater and stood the grater in a cake pan for stability. If the peel starts slipping around and resists being grated, set it aside; once you've finished grating the potatoes, chop the peel and add it to the grated potatoes. Put all of that into a medium bowl. Season generously with salt and pepper, and stir together with your hands.
Add the olive oil and butter to a large nonstick skillet and heat it over a medium flame. Put in half the potatoes. Using a wide spatula, press them flat and round the edges. Scatter the gruyere on top and level it evenly with the spatula. Dot with a little butter here and there. Then put on the rest of the grated potato on top, even it out and press down all around with the spatula. Reduce the heat to medium-low, and sauté for about 6 to 8 minutes - lift up the pancake to make sure it isn't burning on the bottom.
When the bottom is nicely golden brown, it's time to flip the rösti. Everyone has their preferred way of handing this. I slid the whole thing from the skillet onto a cutting board, then inverted the skillet over the pancake and quickly flipped the whole operation—easy! Sauté that side too, about 6 to 8 minutes. Transfer to a platter and slice into wedges to serve.