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Line a 900g loaf tin with a large piece of foil. Press down well into the base and sides of the tin, and leave plenty overhanging the edges. Put the biscuits in a freezer bag and bash to crumbs with a rolling pin. Mix with the melted butter and tip the crumbs into the prepared tin – press firmly into the base to create an even layer. Chill while the filling is made.
Use an electric whisk on a medium setting to beat the condensed milk, cheese and cream together until smooth and thickened – this will take around 2 mins. Add the grated zest and juice, and beat slowly until the cheese mixture forms soft peaks.
Spoon the lime mixture over the biscuit base. Cover lightly with the overhanging foil and seal firmly to keep out as much air as possible. Wrap in a second layer of foil, pop into a large freezer bag and freeze overnight.
To make the cranberry compote, put the cranberries, sugar and 150ml cold water in a saucepan and bring to a simmer. Cook for 5 mins, stirring occasionally, until the cranberries are softened but holding their shape. Mix the arrowroot or cornflour with 2 tbsp cold water until smooth, then stir into the fruit. Cook for another 2 mins, stirring regularly. Leave to cool, then cover and chill. Can be frozen at this stage.
Turn out the frozen cheesecake onto a platter or board and leave to thaw for 30 mins, then carefully peel off the foil. Warm the cranberry compote gently and serve with the sliced semifreddo.