Grate the potatoes onto a clean tea towel (you will need 1 litre/4 cups grated potatoes). Wring out the liquid by wrapping the potatoes in the towel and twisting it over the sink.
In a bowl, combine the potatoes and green onions. Season with salt and pepper.
Heat the butter and oil in two large non-stick skillets over medium heat. Divide the potato mixture into 3 pancakes per skillet, pressing each pancake lightly together. To shape perfectly round rösti, press the mixture into an 8-cm (3/4-inch) cookie cutter. Fry the rösti for 8 minutes over medium heat. Carefully flip the rösti. Continue frying until tender and golden, about 8 minutes. Add butter as needed. Keep warm.
In the top part of a double boiler, off the heat, whisk together the carrot juice, lemon juice and egg yolks. Season with salt and pepper. Place the double-boiler over a pot of simmering water and whisk constantly until the sauce is thick and creamy. Cover with plastic wrap. Set over a saucepan of hot (but not boiling) water.
Melt the butter in a large non-stick skillet over medium heat. Fry the eggs in the butter until the whites are set and the yolks warm, about 5 minutes.
Top each rösti with a fried egg and drizzle with sabayon. Garnish with Parmesan shavings and chives.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.