Peel and finely slice the onions. Peel the carrot and finely chop. Peel the garlic clove and crush or finely chop. Pick the thyme leaves.
Pour 1 tbsp oil into a large pan and warm to a medium-high heat. Sprinkle a pinch of salt and pepper over the pigeon breasts and add them to the hot pan. Fry for 2 mins on each side, till browned on both sides, then scoop out of the pan and transfer to a plate and set aside.
Reduce the heat of the pan to medium. Slide in the chopped onions and carrot along with the garlic, thyme and a pinch of dried chilli flakes (chilli flakes are spicy, so add a small pinch, you can always increase the amount later). Cook for 5 mins, stirring frequently, till slightly softened.
Drain and rinse the borlotti beans. Scatter the beans into the veg in the pan. Pour in the passata. Crumble in the stock cube and add 200ml boiling water. Stir well to combine. Bring back to the boil, then reduce to a simmer and cook for 10 mins.
While the stew is simmering, strip the cavolo nero leaves away from any woody stalks. Finely shred the leaves.
After 10 mins, stir the shredded cavolo nero leaves through the stew till they are tender. Pop the pigeon breasts and any resting juices back into the pan. Warm through for a further 5 mins.
Have a taste of the sauce and add a little more salt and pepper if needed. Spoon the pigeon stew into bowls and serve.