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Bring a deep saucepan of water to the boil (at least 2 litres) and add 3 tbsp white wine vinegar. Lower the heat down to a gentle simmer.
Break the eggs into four separate coffee cups or ramekins. Split the muffins, toast them for a few minutes either side and warm some plates.
Melt the butter in a saucepan and skim any white solids from the surface. Keep the butter warm.
Put the egg yolks, white wine or tarragon vinegar, a pinch of salt and a splash of ice-cold water in a metal or glass bowl that will fit over a small pan. Whisk for a few minutes, then put the bowl over a pan of barely simmering water and whisk continuously until pale and thick, about 3-5 mins.
Remove from the heat and slowly whisk in the melted butter bit by bit until it’s all incorporated and you have a creamy hollandaise. (If it gets too thick, add a splash of water.) Season with a squeeze of lemon juice and a little cayenne pepper. Keep warm until needed.
Swirl the simmering vinegared water briskly to form a vortex and slide in an egg. It will curl round and set to a neat round shape. Cook for 2-3 mins, then remove with a slotted spoon.
Repeat with the other eggs, one at a time, re-swirling the water as you slide in the eggs. Spread some sauce on each muffin, scrunch a slice of ham on top, then top with an egg. Spoon over the remaining hollandaise and serve at once.