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To make the peperonata, heat the oil in a wide, shallow pan. Add the peppers, onions and garlic and cook gently for 15-20 mins. Season, then stir in the rest of the ingredients and continue to simmer gently for 25-30 mins until everything cooks down into a chunky ragout. Turn off the heat. Can be made up to three days ahead, chilled then reheated to serve.
To make the rösti, cover the potatoes with cold water and add a good pinch of salt. Bring to the boil and simmer for 8 mins until half-cooked. When a knife is inserted they should still be raw in the middle. Strain and leave until cool enough to handle.
Peel the potatoes and discard the peel, then coarsely grate into a mixing bowl and season. Heat half the butter in a non-stick frying pan until sizzling, then add the grated potato, spread out evenly and press down with a spatula. Cook over a low-medium heat for 10-15 mins then, once browned, place a plate on top and flip over. Add the rest of the butter to the pan and, once melted, put the rösti back in the pan and repeat until brown on both sides.
While the rösti is cooking, put a large pan of water on to boil and add the vinegar. Once the water is at a rolling boil, crack in the eggs and poach for 3 mins, then remove with a slotted spoon and place on kitchen paper to absorb any excess water. To serve, divide the rösti into four, top with a good spoonful of warm peperonata and an egg, then finish with some torn basil leaves and a drizzle of olive oil.