To begin, place the cooked potatoes into a large bowl. Roughly mash with a fork to keep some texture, rather than aiming for a completely smooth mash. Transition to mixing in the vegetables.
Add the cooked cabbage or sprouts into the bowl with the potatoes. Season with salt and black pepper, then mix until evenly combined. Set aside while preparing the onion.
Heat 1 tbsp dripping or butter in a large frying pan over medium heat. Add the chopped onion and cook until soft and lightly golden. Move on to combining with the vegetable mixture.
Stir the cooked onion into the potato and cabbage mixture. Mix thoroughly to ensure the flavours are evenly distributed. Proceed to shaping the mixture.
Form the mixture into a large round cake or divide into smaller patties for easier flipping. Compact the mixture firmly to help it hold shape. Transition to frying.
In the same frying pan, heat the remaining dripping or butter until hot. Swirl to coat the base evenly. Move on to cooking the bubble and squeak.
Place the cake or patties into the pan. Cook over medium heat for 8–10 minutes until the underside is crisp and golden. Resist moving them too early to prevent breaking. Transition to flipping.
Turn carefully using a wide spatula or plate-over-pan method if cooking one large cake. Cook the second side until crisp and golden, about 8 minutes. Proceed to garnishing.
Serve immediately with fried eggs, cold cuts of roast meat, or simply with a dollop of brown sauce. Presentation tip: cut into wedges if made as one large cake for a rustic table serving.