For the potato rösti, grate the potato into a large bowl lined with a clean tea towel. Squeeze the tea towel to remove the excess liquid from the potato then season with the salt.
Heat the butter and olive oil in a frying pan, add the grated potato and press down. Fry for 4-5 minutes on each side, or until crisp and golden. Remove from the pan and cut into wedges. Arrange the wedges on a serving plate.
For the poached eggs, put the vinegar into a small saucepan and fill up with boiling water. When the water is gently simmering, swirl the water round with a spoon to make a whirlpool in the middle. Crack an egg into the whirlpool and leave for 3-5 minutes, or until the egg has set.
Remove with a slotted spoon and repeat with the other egg.
To serve, place the poached eggs on top of the rösti potato wedges.