For the soda bread, preheat the oven to 180C/350F/Gas 4.
Sift the flour, bicarbonate of soda, cream of tartar and salt into a large bowl. Stir in the sugar until well combined.
Gradually add the buttermilk a little at a time, mixing well until the mixture comes together as a soft dough. (NB: You may not need all of the buttermilk. Take care not to overwork the mixture.)
Dust your hands with flour and knead the dough very lightly in the bowl. Shape the dough into a loose ball and place onto a floured baking tray. Cut a deep cross into the top of the loaf using a sharp knife.
Transfer the soda bread to the middle rack of the oven and bake for 30-35 minutes, or until the bread has risen and is golden-brown and sounds hollow when tapped on the bottom. (Keep the oven on after you remove the bread.)
Meanwhile, for the potato farls, melt the butter in a small pan over a low to medium heat. Add the mashed potatoes and salt and stir well to combine. Fold in the flour until well combined.
Turn the mixture out onto a lightly floured work surface and knead lightly until smooth.
Divide the mixture in half. Roll out each half onto a lightly floured work surface until it forms a circle about the size of a large dinner plate and is about 0.5cm/¼in thick. Cut each circle into quarters.
Heat a large, non-stick frying pan over a medium heat until hot. Add the potato farls, in batches, and fry for 3-4 minutes on each side, or until golden-brown on both sides. Keep warm until ready to serve.
Meanwhile, for the fry-up, arrange the sausages in a roasting tray. Transfer to the oven and roast for 25-30 minutes, or until golden-brown and cooked through.
Place the tomato halves, cut sides facing upwards, into a separate roasting tray with the button mushrooms. Drizzle over one tablespoon of the olive oil and season, to taste, with salt and freshly ground black pepper. Roast in the oven for 10-15 minutes, or until softened.
Meanwhile, heat another tablespoon of olive oil in a large non-stick frying pan over a medium heat. Add the slices of black pudding and fry for 1-2 minutes on each side, or until crisp on both sides. Remove from the pan and keep warm.
Add the bacon rashers to the same pan and dry fry for a 6-8 minutes, or until crisp and golden-brown on both sides.
Meanwhile, heat the remaining two tablespoons of olive oil in a separate frying pan over a high heat. When the oil is hot, crack in the eggs, in batches if necessary, and fry for 1-2 minutes, or until the yolk has just set.
To serve, place two potato farls on one side of each serving plate. Divide the sausages, black pudding, bacon, tomato halves, mushrooms and fried eggs among the serving plates, alongside the potato farls. Slice the soda bread, spread with butter and serve alongside.
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