Mix the butter, salt, icing sugar, almond powder and vanilla powder to obtain a crumble. Then add the eggs and the flour last. Knead gently until the pastry is homogeneous. Keep in the refrigerator (4°C (39°F)) for 24 hours before using.
Mix together the softened butter and icing sugar and slowly add the eggs. Then add the almond powder, custard powder and dark rum (optional). End by adding the cheese cream and pineapple pulp. Put in the refrigerator.
In a saucepan, heat the mango purée and glucose to 30°C (85°F). Then add the sugar mixed with the pectin NH and boil for 2-3 minutes. Pour into 3 cm diameter sphere silicon moulds (déco relief). Put it into the freezer and set aside for the light Caribbean mousse domes.
Prepare Italian meringue: in a saucepan, heat the sugar and water to 121°C (250°F), pour over the egg whites and beat until completely cooled. Add the melted prepared gelatine to the Caribbean cocktail purée, then add the cooled Italian meringue and whipped cream last. Open the 4 cm sphere "déco relief" silicon moulds, pipe in a little mousse, insert the frozen mango coulis dome, close the silicon mould and finish piping the rest of the mousse. Put everything in the freezer. If you do not have a "déco relief" silicon mould, you can use Flexipan (fleximold) silicon moulds with the same diameters and stick the halves together.
Roll out the short pastry dough until it is 3 mm thick. Cut out circles of 7 cm diameter and put them in the refrigerator for about one hour. Use a #7 nozzle to pipe each circle with pineapple almond cream (about 25 g (0.8 oz)), line the inside with some finely diced fresh pineapple (about 20 g (0.7 oz)) and then bake in a convection oven for about 25 minutes at 160°C (325°F).
Remove the light Caribbean cocktail mousse spheres from the moulds and then coat with a green-tinged neutral glaze using a wooden skewer.
Mix small pieces of pineapple with some neutral glaze and place them on the cooked tartlets bases. Then place a frozen sphere in the centre. Decorate with a small cube of coconut marshmallow and a dash of lime zest.