- Grate the potatoes on a clean cloth (you will need about 2 cups/500 ml). Wring the potatoes dry by twisting the cloth over the sink.
- In a bowl, combine the potatoes with the green onion. Season with salt and pepper.
- In a large non-stick skillet, heat the butter and oil over medium heat. Divide the potato mixture into four patties and press down lightly (see note). Cook for 8 minutes. Carefully turn the patties over and cook for 8 minutes or until the potatoes are tender and golden brown. Add butter, as needed. Set aside in a warm oven, if necessary.
- Meanwhile, over a double boiler off the heat, combine the lemon juice, water and egg yolks with a whisk.
- Place over simmering water and whisk until the mixture is hot and foamy. Remove from the heat. Drizzle in the butter, whisking constantly. Season with salt and pepper. It is important not to bring the sauce to a boil.
- In a pot of salted simmering water, add the vinegar. Break the eggs, one at a time, into a ramekin and slip into the water. Poach for about 3 minutes or until the desired doneness. The yolks should still be runny. Drain the poached eggs on a plate lined with paper towels.
- Divide the rösti among four warm plates. Top with the eggs, drizzle with the hollandaise sauce and garnish with the lobster claws.
If you want perfectly round rösti, press the mixture into a 3 ¼-inch (8 cm) cookie cutter. If you host a brunch, you can cook and shell the lobster in advance and prepare and cook the rösti. You will only have to warm them in the oven at the last minute and prepare the hollandaise sauce.