For the soup, remove the dark gills from the mushrooms and discard them. Wash and dry the mushrooms, then cut into 2mm slices.
Heat the olive oil in a large heavy-based pan, add the shallots and cook until just coloured. Add the garlic and cook until soft, then add the mushrooms.
Season with salt and freshly ground black pepper to taste, and cook for 5-8 minutes or until the mushrooms are golden.
Add the lemon juice and the stock. Bring to a rapid simmer and then turn down to a low heat to gently simmer for 15 minutes.
Check the seasoning again, adding more salt and freshly ground black pepper, to taste, if necessary.
Ladle the soup into a blender or food processor and allow to cool for two minutes. Blend until smooth, then strain through a fine sieve back into a clean pan.
To finish, add the cream and stir it in over a medium heat - you can also stir in cubed butter for a silkier finish. Keep warm.
For the red mullet and mushrooms, heat a frying pan until hot then add three tablespoons of olive oil, the wild mushrooms and a pinch of salt.
Stir-fry the mushrooms until cooked through, then remove them with a slotted spoon and transfer onto a plate lined with kitchen paper.
Season both sides of the red mullet with salt and freshly ground black pepper.
Heat a second frying pan until hot then add the remaining olive oil and the red mullet skin-side down.
Give the pan a little shake to ensure that the fish doesn't stick. The fish may curl up, but do not press it down as this will zip back the skin - leave the fish be and the fillets should flatten out naturally.
Cook for 1½ minutes on the skin side, then turn over and cook for a further minute, or until completely cooked through.
Remove the fish from the pan and transfer to a plate lined with kitchen paper.
For the shallot and herb crème fraîche, place the crème fraîche into a small bowl and add the shallots and chopped herbs. Add salt, freshly ground black and lemon juice to taste and stir to combine.
To serve, put a small mound of the cooked wild mushrooms in the middle of four wide-rimmed soup bowls. Place a fillet of red mullet on top of each serving and pour the soup around. Spoon the crème fraîche mixture on top of the fish, so it melts into the soup. Drizzle extra virgin olive oil around to taste.