For the salad, place the beetroot into a saucepan and cover with water. Bring to the boil then reduce the heat and simmer for 20-25 minutes until the beetroot is tender. Drain and cut the beetroot into quarters lengthways.
Heat a frying pan until hot then add the oil and shallots and cook gently for two minutes until just softened.
Add the mushrooms and fry for two more minutes until the mushrooms are just tender. Tip the mixture into a large bowl then add the chestnuts, celery and salad leaves.
Whisk the mustard, vinegar and extra virgin olive oil together and season with salt and freshly ground black pepper.
Place the vegetable oil into deep-sided saucepan and heat until a small cube of bread sizzles and turns golden when dropped into it. (CAUTION: hot oil can be dangerous. Do not leave unattended.) Alternatively, heat a deep-fat fryer to 190C/375F.
Carefully place the potato chips into the hot oil and deep fry for 1-2 minutes until golden-brown. Remove from the oil with a slotted spoon, drain on kitchen paper and season with salt and freshly ground black pepper.
For the pigeon, preheat the oven to 200C/400F/Gas 6. Season the pigeon breast with salt and freshly ground black pepper.
Heat an ovenproof frying pan until hot then add the olive oil and butter. Cook the pigeon breasts, skin side down, for one minute until golden-brown then turn and cook for another minute. Transfer the pan to the oven and roast for three minutes.
Remove the pan from the oven and leave to rest for five minutes, basting occasionally with the pan juices.
To serve, toss the beetroot quarters into the salad and dress with the dressing. Carve the pigeon into thick slices then add to the salad. At the very last minute, add the game chips and toss lightly to combine.