Preheat the oven to 170C/160C Fan/Gas 3. Grease two baking trays and line with parchment.
Cream the butter and sugar together in a bowl until pale and fluffy. Stir in the vanilla extract.
Sift and fold the flour and cornflour gradually until it binds together. Gently knead until just smooth (be careful not to over-work it). Wrap in cling film and place in the fridge for 30 minutes.
Flour a work surface and roll out the dough to 5mm/¼in thick. Use a large cookie cutter to cut into 48 circles. Transfer to the prepared baking trays and bake for 15–20 minutes, or until pale golden-brown.
Sprinkle with the remaining caster sugar and leave to cool for 10 minutes. Carefully transfer to a wire rack.
For the buttercream, put the butter, half the milk and all the vanilla extract in a large mixing bowl. Beat until smooth using an electric mixer. Sift over the icing sugar and, with the mixer on a low speed, continue mixing until smooth. Add a little more milk if required.
For the jam, set aside a small handful of strawberries and mash the rest to a rough pulp. Pour the pulp into a wide, heavy-bottomed pan. Add the jam sugar and lemon juice and bring to the boil. Add the remaining strawberries and boil for about 5 minutes, see the tip section for how to check if the jam has set. Remove from the heat and skim off the pink scum. Leave to cool.
Put a layer of cream, then a layer of jam onto a shortbread and use another round to make a ‘sandwich’. Fan the sliced strawberries on top. Repeat the process with the remaining shortbread, buttercream, jam and strawberries to make 24 sandwiches.