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Mix together the crème fraîche and Dijon mustard, and set aside. Toss the salsify in the lemon juice to stop them discolouring.
Heat the vegetable oil in a large non-stick frying pan and fry the salsify for 2-3 mins, then add the butter. Turn the heat down and cook gently for 10-12 mins, or until the salsify is soft and golden. Keep the salsify warm while you poach the eggs.
Top the crumpets with a generous pinch of the grated mature cheddar and grill until melted and bubbling. Sprinkle the salsify with a pinch of grated nutmeg and season.
Pile the salsify over the crumpets, then top with a poached egg each and the mustardy crème fraîche. Serve with a few chopped chives, some black pepper and a drizzle of butter from the pan.