Heat a splash of oil in a saucepan over a low heat. Add the broccoli and season well. Cook for 20 minutes until the broccoli is soft and caramelized. Remove from the pan and chop into small pieces.
Combine the broccoli, Cheddar, potatoes, eggs and one-third of the breadcrumbs. Season well, then combine and shape into patties.
Place the remaining breadcrumbs in a shallow bowl. Toss the broccoli and potato cakes in the breadcrumbs to coat.
Preheat the oven to Gas Mark 1, 140°C (375°F). Place a large baking tray in the oven to keep warm.
Heat a thin layer of oil in a large, heavy-based pan over a medium heat. Working in batches to avoid crowding the pan, cook the broccoli and potato cakes for 2-3 minutes per side until browned. Transfer the finished cakes to the tray in the oven to keep warm while you cook the remainder.