Tumble the whole potatoes into a pan. Pour over enough boiling water to just cover them. Bring to the boil, cover and simmer for 10 mins till just tender. Drain and rinse with cold water.
Pat the potatoes dry with kitchen paper. Coarsely grate them into a bowl (no need to peel them). Roughly chop the dill. Add to the bowl with a generous pinch of salt and pepper. Mix together. Set your oven to its lowest setting.
Heat 1 tbsp olive oil and 1 tbsp butter in a large, non-stick frying pan. Add four heaps of potato to the pan, gently shaping them into rounds with a spoon. Press down lightly to flatten. Cook for 5-6 mins over a low heat till golden underneath.
Loosen the rösti with a spatula and lift them out of the pan. Add another 1 tbsp butter and melt it. Return the rösti to the pan, uncooked side down. Cook for a 5-6 mins till golden underneath. Place on a plate. Put in the bottom of the oven to keep warm.
Peel and slice the shallots. Heat 1 tbsp olive oil in a small pan. Add the shallots. Cook gently for 5 mins. Peel the carrots. Slice into thin sticks or use a julienne cutter to make thin strips. Add to the shallots. Cook for 1 min.
Finely grate the zest from one half of the lemon. Juice that half. Slice the other half into wedges. Stir the zest and juice into the veg. Take off the heat. Season. Pop on a lid to keep them warm.
Heat 1 tbsp olive oil in the frying pan. Season the pigeon. Add to the pan. Cook for 4 mins, turning once, till firm and brown. Add the remaining butter to the pan. Let it sizzle. Remove from the heat. Rest for 5 mins. Serve the pigeon with the röstis, veg, baby leaf salad and lemon wedges.