For the charred leek oil, heat a griddle. When hot, add the leeks cut-side down and cook until charred. Turn the leeks over and cook on the other side until soft. Place in a bowl with the oil and leave in a warm place (ideally around 50C/122F) for 4-5 hours. Then leave overnight to infuse. Strain the oil before using.
For the leek mayonnaise, in a food processor, blend the egg yolks and mustard to combine. With the motor running add the leek oil in a thin steady stream until the mixture emulsifies to form a mayonnaise. Add the white wine vinegar and pulse to combine. Season with salt, pepper and a little lemon juice, to taste. Store in the fridge until ready to serve.
For the leeks and girolles, season the whole leek with salt and pepper and place on a hot griddle until charred all over and soft. Transfer to a baking tray and cover with cling film to keep warm.
Season the baguette slices with salt and pepper and brush with olive oil. Toast both sides on the griddle to make croûtons.
Warm the remaining leek oil in a small pan set over a low heat. Gently lower the egg yolks into the oil and poach very gently for about six minutes.
Heat the butter in a frying pan and, when foaming, fry the girolles until soft. Add the vegetable stock, then turn off the heat and stir a little vinaigrette into the mixture.
To serve, cut the warm leek in half and remove the blackened outer layer. In shallow bowls, toss the leek in the remaining vinaigrette. Place the egg yolks on top, followed by the mushrooms and a spoonful of leek mayonnaise. If desired spoon over some of the cooking liquor from the mushrooms.
Garnish with the toasted baguette and herbs.