Boil the potatoes in a pan of salted water for 15–20 minutes until tender. Meanwhile, finely shred the cooked Brussels sprouts.
Drain the potatoes and mash thoroughly using a potato masher. Return to the pan and stir over a low heat for 1-2 minutes to dry out. Add the butter to melt and mix well, then take off the heat.
While the potatoes are still warm, mix in the shredded Brussels sprouts. Season well with salt and pepper.
Tip the flour onto a plate and season with a pinch each of salt and pepper. Shape the potato and sprout mixture into 10–12 patties and lightly coat with the seasoned flour.
Heat 5mm (¼in) olive oil in a large frying pan. In batches, fry the cakes for 2 minutes on each side, until golden. Drain on kitchen paper, sprinkle with a little salt. Serve with bacon and egg, if you like.
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If serving vulnerable groups, elderly people, toddlers, pregnant women and people who are unwell, cook eggs until the whites and yolks are solid.