Boil the potatoes in salted water until just tender but not fully cooked, about 12 minutes. Cool completely and refrigerate until firm.
Coarsely grate the potatoes. Toss with salt and pepper.
Cook the Rösti:
Melt 2 tablespoons butter in a nonstick skillet over medium heat. Add half the potatoes, sprinkle over the cheese, and cover with remaining potatoes.
Press into an even cake. Cook until the bottom is deeply golden, 8 to 10 minutes. Invert onto a plate, add remaining butter, and slide the rösti back into the skillet.
Cook the second side until crisp and the cheese is melted, 8 to 10 minutes.
Top and Serve:
Fry the eggs in a separate skillet until the whites are set and the yolks remain runny.
Cut the rösti into wedges and top each serving with a fried egg, chives, and salad.
Notes
Cool the potatoes before grating.
Press the cake firmly but do not stir it while cooking.