Place the whole eggs, unbroken, with the truffle and basmati into a bowl and leave for at least 24 hours for the truffle to infuse through the shells into the eggs. (The basmati is to absorb any moisture. It’s not cooked in this recipe, but can be kept and cooked later.)
Season the halibut with a little salt.
Heat a frying pan until medium hot, add a little butter and the olive oil. Cook the halibut, skin side down, over a medium heat for four minutes until the flesh has turned opaque.
Flip over and cook for another minute then remove from the heat and rest in the pan for a couple of minutes.
Bring a pan of salted water to the boil and add the vinegar. Whisk the water to create a whirlpool and once settled, crack an egg in. Simmer for 2-3 minutes, remove the poached egg carefully with a slotted spoon and place on a plate to drain. Repeat the process with the remaining eggs.
Heat another frying pan with the remaining butter and heat until foaming. Add the cavalo nero and 75ml/3floz water and cook for 2-3 minutes until just softened and the water and butter emulsified.
For the hollandaise, place the butter in a saucepan and bring to a simmer.
Place the lemon juice and vinegar in a separate small pan and heat until just boiling.
Place the egg yolks in a food processor with a pinch of salt and turn on.
With the motor running, add the hot lemon and vinegar. Then slowly add the melted butter until it is all incorporated. Remove from the processor and season with a little salt and black pepper.
To serve, lift the skin off of the halibut and discard. Divide the cavalo nero between four plates and top with a fillet of halibut. Place the poached eggs on top, then spoon the hollandaise over and finish with a grating of fresh truffle.