Bring a big pan of water to a simmer, carefully lower in the eggs and cook for 8 minutes. The eggs should be slightly runny in the middle. Run under a cold tap until they are completely cold and carefully peel off the shells.
Melt the butter in a pan and cook the shallot until soft. Mix with the other ingredients, seasoning well.
In a piece of cling film, pat out a piece of the sausage-meat mixture large enough to envelope an egg. Put the egg in the middle and, using the cling film, fold it around the egg so it is completely covered. Remove the cling film and mould with your hands so there is an even thickness all the way around. Repeat with all the other eggs and put into the fridge to firm up.
For the coating; whisk the eggs and milk with a pinch of salt. Dust the sausage-coated eggs in the flour and shake off the excess, then dip into the egg mix and then through the breadcrumbs. Dip back through the eggs and into the crumb mix again. Place into the fridge to firm up.
Heat a fat fryer to 185°C (or warm up vegetable oil in a pan using a probe to gauge the temperature accurately) and the oven to 200°C. Brown the Scotch eggs for 2 minutes in the fryer then place in the oven for 10 minutes to warm through.
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