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Heat oven to 180C/160C fan/gas 4 and grease and line a 20cm square shallow cake tin with baking parchment. Put the apricots and conserve in a small pan with 4 tbsp water, simmer over a medium heat until thick, then mash the apricots a little with a fork and set aside to cool.
Tip the butter, sugar and vanilla into a large bowl, and beat with an electric whisk until pale and fluffy. Add the flour and ground rice, using a wooden spoon and then your hands to bring the mixture together as a dough. Divide into 2 pieces.
Use your fingers to spread 1 piece of shortbread dough over the bottom of the tin. Spread the apricots over the top, leaving a small border around the edge. Roll out the remaining dough to a 20cm square and place on top of the apricot filling. Use your finger to make an indent all around the edges of the shortbread, then prick the top all over with a fork. Bake for 25-30 mins until the edges are just starting to turn golden. Leave to cool in the tin, then sprinkle with caster sugar and cut into 16 squares.