For the white wine sauce, heat a heavy-based pan over a medium heat, then add the oil.
Add the shallots and fry gently for 3-4 minutes, or until softened.
Add the vermouth and fish stock and bring the mixture to the boil. Reduce the heat until the mixture is simmering, then continue to simmer for 15-20 minutes, or until the volume of liquid has reduced and the sauce resembles syrup.
Add the cream, then return the mixture to the boil. Reduce the heat until the mixture is simmering, then simmer again for a further 6-8 minutes, or until the volume of liquid has reduced by about half and the sauce has thickened. Season, to taste, with salt and freshly ground black pepper.
Strain the mixture through a fine sieve into a clean bowl. Stir in the mustard and chopped dill until well combined.
Meanwhile, for the scallops, preheat the oven to 200C/400F/Gas 6.
Slip a thin, sharp knife between the scallop shells, then twist to prise the scallop shells apart.
Cut through the large white muscle that attaches the scallop to one of the shells, then scoop out the meat using a spoon.
Wash the scallop well, then separate the white meat from the outer skirt and roe (the skirt and roe can be frozen and used in another dish).
Rinse the scallops and pat them dry with kitchen paper. Rinse out four of the shells and pat them dry (the other shells can be discarded). Cut each scallop in half horizontally to form two thinner scallops.
Heat the oil in a frying pan over a low heat. Add the carrot and leek and fry for 3-4 minutes, or until softened but not coloured. Set aside to cool.
Spoon the cooled carrot and leek mixture into the four scallop shells, then arrange four pieces of scallop on top of each in a circle. Spoon over three tablespoons of the white wine sauce, then cover each with the remaining shell.
Roll out the puff pastry onto a lightly floured work surface to a 0.5cm/½in thickness. Cut it into four thin strips.
Wrap one of the pastry strips around each scallop and seal the edges by pressing them together using your fingers (this will secure the two shells together).
Brush the pastry strip all over with the beaten egg yolk. Place all of the scallops onto a baking tray, then transfer to the oven and cook for 8-10 minutes, or until the pastry has risen and is golden-brown and the scallops are just cooked through.
To serve, pile the seaweed or rock salt into the centre of four serving plates (if using rock salt, sprinkle it with a little water to stop the scallop from moving about). Cut the pastry seals of the scallops using a sharp knife, but keep the scallops closed. Place one onto each pile of seaweed or rock salt.