1. Preheat the oven to 220ºC gas mark 7. Make the rum butter first. Cream the butter until smooth then beat in the icing sugar, brown sugar and mandarin zest and continue beating until pale and fluffy. Beat in the mandarin juice and rum a little at a time. Transfer to a dish and chill until needed.
2. Roll out the pastry to a rectangle about 30x40cm. Cut out eight 15cm rounds from the pastry. Place a large tablespoon of the mincemeat in the centre of each round and brush the pastry edges with water. Lift the pastry up around the filling and pinch it together to seal. Turn the Eccles cakes over so that the edges are underneath and arrange on a damp baking sheet. Gently press down with a rolling pin to form flat cakes.
3. Cut 2 small slits in the top of each cake. Brush with egg white and sprinkle with the sugar. Bake the Eccles cakes for 15 minutes then turn the oven down to 180ºC gas mark 4, and cook for a further 15-20 minutes until golden and sticky. Leave to cool on a rack and serve warm or cold with the rum butter.