Place the pear halves in a pan with the Calvados and water. Bring to the boil then remove from the heat, cover and allow to cool for at least 2 hours.
For the pastry case, rub the butter into the flour until the mixture resembles fine breadcrumbs. Add the water and mix to a firm dough. Roll out the pastry and line 6x10cm loose-based flan tins.
For the frangipane, cream together the butter and sugar until pale and fluffy. Lightly beat the eggs and add a little at a time, beating well after each addition. Add the almonds and flour and gently fold in.
Spread 2 heaped teaspoonfuls of the jam over the base of each tartlet then spoon the frangipane into the cases so that they are two thirds full. Smooth the surface and place two pear halves side by side on the top of each tartlet.
Bake at 180 °C/350F/gas 4 for 25-30 minutes until the frangipane is set and golden brown.
Remove from the tins, brush with the remaining warmed apricot jam and dust with icing sugar before serving.