To make the tarts, dust a work-surface lightly with flour and roll the puff pastry a little thinner. Chill to let it rest about 20 minutes.
With a very sharp knife, cut dough into 3-inch squares. Fold each square in half diagonally, forming an isosceles triangle. Lay triangle on a flat surface with the base closest to you. Starting at the base, cut a 1/4-inch wide band along each side of the triangle, stopping just before you reach the point.
On a sheet pan lined with parchment paper, unfold the triangle. There should be a band around the entire square that is attached only at 2 opposing corners. Paint the band with water. Grab the band at the 2 detached corners and cross them over each other, creating a rim around the tart shell and a decorative loop at the 2 corners that are still attached. Press the band into the dough.
Prick the bottom then chill again, for 20 minutes.
Preheat oven to 425 degrees F.
Bake for 20 minutes in the top 1/3 of the oven until golden brown. Cool on the pan.
Knock down the centers of the tart shells if needed.
Once the tart shells have cooled, put 1 tablespoonful of the chocolate ganache into each of them. Cover the middle with sections of pineapple rings and place half cherries at the corners. Warm the apricot jam and water together, and brush it on to glaze the tarts.
Let the glaze set up, and then serve as soon as possible.
Place the chopped chocolate in a bowl.
In a saucepan, heat the cream until boiling. Pour it over the chocolate and whisk well. Whisk in the rum and strain it into a bowl. Cover and chill overnight. This next day carefully whip it until stiff.
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