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Heat oven to 160C/140C fan/gas 3. Put the parsnips in a roasting tin with the oil and roast for 1 hr 30 mins. Remove from the oven and cut into chunks.
Meanwhile, make the béchamel. Melt the butter in a saucepan over a low heat, then add the flour and whisk. Remove from the heat and add the milk gradually, whisking after each addition until combined. Return to a low heat and cook, stirring, until thick. Stir in the remaining ingredients, except the Gruyère, add 1/2 tsp salt and take off the heat. Add the Gruyère and mix well.
Once the parsnips are cool enough to handle, purée in a food processor and set aside. Heat the butter in a pan. Add the shallots and cook for 10 mins or until soft and golden. Tip into a large bowl with the parsnip purée, béchamel and horseradish. Season to taste, cover and cool. When almost cold, spoon into a piping bag and pipe into 3cm-thick lengths onto a baking parchment-lined tray. Freeze for 1 hr until firm but not completely frozen. Remove from the freezer and cut into 5cm lengths. Coat the croquettes in the flour, dip in the egg, roll in the panko breadcrumbs, then place on a parchment-lined tray. Chill until needed. Can be made up to this stage up to 24 hrs in advance.
To fry your croquettes, use a deep-fat fryer or half-fill a heavy-based medium saucepan with vegetable oil and heat to 180C. Cook in small batches for 2-3 mins until golden and cooked through. Lift the croquettes from the oil with a slotted spoon and drain on kitchen paper. Put on a warmed plate and cover with foil until ready to serve.