To make the BBQ cabbage, separate the green outer leaves from the heart of the cabbage. Blanch the outer leaves in a pan of boiling water for 1–2 minutes, then drain and squeeze out as much liquid as possible. Set aside the cabbage heart until later.
Dry these leaves in a dehydrator, or in the oven at the lowest setting, for 3–6 hours or until completely dried out. Set aside a couple of leaves.
Preheat the oven to 170C/150C Fan/Gas 3½. Toast the nori sheets on a baking tray in the oven for 10 minutes. Combine the toasted nori sheets and dehydrated cabbage leaves in a small blender or food processor and blend to a fine powder.
Halve the heart of the cabbage and place in salted water for 1–3 hours.
Heat the oil in a deep-fat fryer to 160C (CAUTION: hot oil can be dangerous. Do not leave unattended).
Deep-fry the reserved dehydrated cabbage leaves for 1 minute. Remove the ‘crisps’ from the oil and drain on kitchen paper. Season with a pinch of salt and break into random pieces.
Lift the cabbage heart from the salted water, drain and cut each half into 2–3 wedges. Heat a barbecue or grill to high, brush the cut side of the wedges with oil and char until tender.
To make the smoked eel cream, heat the vegetable oil in a pan and add the eel skin and bones. Fry gently for about 10 minutes until the flavour and fats are released. Add the cream and bring to the boil, then remove from the heat and infuse for 20 minutes. Strain the cream and season it with the white soy sauce. Allow to cool to room temperature.
Preheat the oven to 140C/120C Fan/Gas 1. Warm the eel in the oven for 6 minutes. Remove from the oven and dress with the fermented dulse, if using.
Heat the oil in a frying pan and fry the eggs to your liking. Add a spoonful of the eel cream to each plate.
Fan out the wedges of cabbage and place one on each plate. Scatter the eel amongst the cabbage leaves and season with lemon juice.
Top each plate with a fried egg, dust heavily with the cabbage/nori powder and garnish with the ‘crisps’.