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Put the egg yolk, butter, vanilla, sugar and salt in a food processor, and pulse until creamy and soft. Add the flour and pulse until the mixture comes together in clumps – don’t overwork it. Tip onto a lightly floured surface and squish the dough together. Split into 2 pieces, one slightly larger than the other, then shape into smooth discs. Wrap in cling film and chill for 30 mins.
Meanwhile, make the filling. Put the berries and sugar in a wide pan and cook for 5 mins until syrupy and the gooseberries are soft but not bursting. Drain in a colander over a bowl and leave to cool. Mix together the cinnamon and extra sugar, and set aside.
Heat oven to 200C/180C fan/gas 6. Line a 23cm fluted tart tin with the larger piece of pastry (see above right). Prick the base several times with a fork, chill until firm, then line with foil and fill with baking beans. Bake on a baking sheet for 15 mins. Remove the foil and beans, and bake for a further 10 mins or until the bottom of the pastry is golden and feels sandy. Roll the second pastry disc to roughly the size of the tart and cut into 5cm squares.
Scatter the semolina or almonds over the pastry base (this will help to prevent a soggy bottom). Top with the fruit and drizzle with 2 tbsp of the syrup. Space the pastry squares over the tart, brush with egg white, then scatter the pie with most of the reserved cinnamon sugar. Wrap only the edge of the pie with a collar of foil to protect it from overcooking, then bake for 30 mins until golden and crisp. Scatter with more spiced sugar and serve warm with thick cream and the fruity pink syrup in a jug for pouring.