Brush the pigeon breasts with the oil and place in a hot pan over a medium heat. Cook for 2-3 minutes, turning once. Remove from the heat and leave to rest.
Heat a griddle pan until smoking. Add the leeks, the oil and the honey and cook for five minutes, turning occasionally to evenly brown.
For the dressing, mix the red wine vinegar and olive oil in a bowl.
To serve, place the leeks in the centre of a plate. Slice the pigeon breast and place on top, drizzled with the dressing.