Put the roast potatoes, carrots and grated cheese into a large mixing bowl. Season with the salt and pepper. Use a potato masher to mash everything together well then add the cabbage and mix through. Divide the mixture into four even portions and shape each one with your hands to make a 2 centimetre thick patty - take the time to neaten up the edges.
Put the frying pan back on the heat and add 1 tablespoon of vegetable oil and the butter. Heat the oil then cook one or two patties at a time, depending on the size of your pan. Cook for 2 minutes each side, or until crispy and deep golden brown, then drain on kitchen paper while you repeat with the rest. The bubble and squeak patties are delicate so take care when turning them and removing from the pan.
Heat a small amount of vegetable oil in a large frying pan over a medium heat. Cook the bacon rashers in batches until crispy.
Serve one per person, topped with three rashers of crispy bacon and a freshly poached egg.