Cook the potatoes in a covered pan of boiling salted water for 15-20 minutes until tender.
Meanwhile, heat a knob of the butter and one tablespoon of water in a heavy-based pan with a lid, over a high heat. When the butter has melted and formed an emulsion, add the scallions and cabbage with a pinch of salt. Cover, shake vigorously and cook over a high heat for 1 minute. Shake the pan again and cook for another minute, then season with pepper.
Drain the potatoes and mash until smooth, then beat in the remaining butter. Fold in the cabbage mixture. Shape the mixture into four balls, dust with flour and press into neat patties.
Heat a thin film of olive oil in a heavy-based frying pan and add the patties, then cook for 3-4 minutes on each side until golden brown.
To make the poached eggs, bring a large pan of water to the boil. Add the vinegar and season with salt and keep at a very gentle simmer. Break the eggs into the water and simmer for 3-4 minutes until just cooked but still soft on the inside. Remove with a slotted spoon and drain well on kitchen paper, trimming away any ragged edges.
To make the hollandaise sauce, place the vinegar and egg yolks in a food processor with a pinch of salt. Blend until just combined.
Gently heat the butter in a heavy-based pan until melted and just beginning to foam. Turn on the food processor and with the motor running at medium speed; pour in the melted butter in a thin, steady stream through the feeder tube. Continue to blitz for another 5 seconds and pour back into the pan but do not return to the heat. Allow the heat from the pan to finish thickening the sauce as you stir it gently for another minute before serving. Season to taste with salt.