For the crust:
Put the flour, almond powder, salt, sugar, almond extract, and butter in the food processor and pulse until the texture resembles coarse sand.
Add enough water to form a ball of dough.
Wrap the dough in plastic wrap and refrigerate for 1 hour.

Roll out the dough on a floured surface and fill a 8″ (20 cm) diameter buttered tart mold. Reserve in the refrigerator.

Using an electric mixer, cream the butter and sugar until smooth and creamy.
While mixing, incorporate the almond powder, then the flour and cornstarch, then the eggs and rum. Mix well.
Refrigerate the preparation for at least two hours.

Preheat the oven to 350 F (180 C).
Pour the preparation into the tart shell and smooth the top.
Arrange the pear halves on top, pressing them in slightly.
Place the tart on the middle rack and bake for 50 to 60 minutes.
Let cool on a rack before dusting with powdered sugar.