For the deep-fried egg, bring a large saucepan of water to the boil, add the vinegar and a pinch of salt. Boil 4 of the eggs for 5 minutes. Remove the eggs from the saucepan using a slotted spoon, and leave to chill in a bowl of ice-cold water.
Crack the remaining egg into a bowl. Whisk with a pinch of salt and pepper.
Peel the boiled eggs. Prepare a shallow plate of flour and a separate plate of breadcrumbs. Dredge the boiled eggs in the flour, then dip into the beaten egg and roll in the breadcrumbs. Set aside on a tray in the fridge until needed.
Heat the oil in a deep-fat fryer to 180C/350F. Alternatively, pour the oil into a deep, heavy-based pan to a depth of 2cm/¾in and bring to the same temperature on the hob. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Fry the eggs until golden-brown. Remove with a slotted spoon, drain on kitchen paper and season with salt.
Heat a frying pan, add the butter and olive oil. Add the shallot and gently fry for 3 minutes, until just softened. Add the garlic and mushrooms and cook until the mushrooms have released their liquid and are tender. Season with salt and pepper.
Heat a clean frying pan. Lay the brioche in the hot pan and toast each side until golden-brown.
For the mayonnaise, put the egg yolks, vinegar and mustard in a food processor and blend until pale and creamy.
With the motor running, pour in a steady stream of oil until the mayonnaise is thick (you may not need all the oil). Just before serving add the herbs and blend.
To serve, cut a hole in the slices of brioche and sit the eggs in the holes. Dress the plate with mushrooms and mayonnaise.