To make the hollandaise sauce, put the beetroot purée, vinegar, egg yolks and salt into a food processor and blitz until smooth. Then slowly add the melted butter. Tip into a bowl and place over a pan of barely simmering water. Whisk until thickened slightly, remove from the heat and keep warm.
Grill the bacon rashers until crisp.Season both sides of the pigeon breasts with salt. Heat the olive oil and butter in a frying pan and fry the pigeon breasts, skin side down, until golden in colour which should take 2 minutes.
Turn the breasts over and cook for a further1-2 minutes, basting the breasts with the oil and butter. Remove the breasts from the pan and allow to rest for 5 minutes.
Meanwhile bring a large saucepan of salted water to the boil and add the vinegar. Reduce the heat until the water is barely simmering and then poach the eggs, two at a time until just cooked.
Place a rasher of bacon, cut in half, on each muffin half and top with 2 sliced pigeon breasts. Top with a poached egg and finally spoon the beetroot hollandaise over the egg. Sprinkle with some chopped chives. Repeat with the remaining muffins. Serve 2 muffin halves per person.