Preheat the oven to 180C/350F/Gas 4.
For the burgers, place the turkey, Brussels sprouts and egg into the bowl of a food processor and blend to combine.
Season well with salt and freshly ground black pepper.
Dust a little flour onto your hands and shape the mixture into four patties. Add more flour to your hands if necessary.
Heat the butter and oil in a frying pan over a medium heat.
Add the burgers to the pan and fry for one minute on each side.
Place the burgers onto a baking sheet, transfer to the oven and bake for about five minutes, or until cooked through.
For the relish, place all the relish ingredients into a small frying pan and cook gently for about eight minutes, until the cranberries have softened.
For the poached eggs, add the vinegar to a small frying pan filled with simmering salted water.
Crack the eggs, one at time, carefully into the pan - take care not to break the yolk.
Place a lid on the pan and cook for two minutes, or until cooked to your liking.
Remove the eggs with a slotted spoon and drain on kitchen paper.
Serve two burgers on each plate with a poached egg and a drizzle of the relish around each serving.