To make the filling, put the apricot pieces in a medium sized bown. Bring the water to a boil and pour over the apricots along with the white wine and the cointreau. Cover and set aside to soak for at least 30 minutes up to an hour.
Transfer the apricots and their soaking water to a saucepan and add the cinnamon stick and white and brown sugars. Bring to a boil, then reduce the heat and simmer, stirring occasionally until the mixture has a thick and jam like consistency, anywhere from 45 minutes to an hour. Set aside to cool. Remove cinnamon stick.
Pre-heat the oven to 205*C/425*F. Unroll the puff pastry. Cut out five or six 4 inch circles from it with a sharp cutter. Take a smaller 3 inch cuter and make another light indentation in the centre of the circle. (there should be a 1/2 inch border all around) Place on a baking sheet lined with parchment paper. Sprinkle the border with the demerara sugar and then dust with cinnamon lightly.
Spoon heaping TBS of the filling into the centre of the pastry, spreading it right up to the border indentation. Do not be tempted to overfill as the mixture will spread upon heating in the oven. Sprinkle the toasted almonds over top of each.
Bake for 15 to 20 minutes in the pre-heated oven until the filling is bubbly and the pastry is lightly browned and crisp. Remove from the oven and immediately remove from the pan with a spatula to a wire rack to cool completely.
Melt the white chocolate in the microwave for 20 to 30 seconds until quite soft and melted. Take a fork and drizzle the white chocolate decoratively across the tarts. Let cool completely before serving.