With your hands, squeeze out as much liquid as possible from the grated potatoes. Pre-heat the grill.
Heat the oil and butter in a large non-stick frying pan, add the leeks and cook for 2 minutes. Stir in the potato, carrot and lightly spread out the mixture in the pan. Fry over a medium heat for 10 minutes until golden underneath.
Place the pan under the grill and cook for 5 minutes or until the rosti is golden on top. Remove from grill.
Make 4 indentations in the top of the rosti. Crack the eggs into the indentations, grind over a little black pepper and scatter over the coriander and chilli. Cover with a lid or baking sheet, place back over a medium heat and cook over for 4-5 minutes or until the eggs are cooked to your liking. Serve straight away.