Preheat oven to 350 degrees F (175 degrees C).

Prepare the dough: In a large mixing bowl, whisk together flour, baking powder, and sugar.

Add softened butter to the flour mixture in small slices, mixing until combined.

Beat in the egg and add heavy cream, blending until a dough forms.

Gather the dough into a ball, cover, and chill in the refrigerator for at least 2 hours.

Prepare the filling: Beat almond paste until softened.

Add rum extract, vanilla extract, and sweetened condensed milk. Blend on high speed for 3-5 minutes.

Gradually add in confectioners' sugar.

Gently fold in blanched almond halves and egg whites.

Set the filling mixture aside.

Press 3/4 of the chilled dough into a lightly greased 9-inch pie pan.

Evenly spread the almond filling over the dough.

On a floured board, roll the remaining dough to about 1/8 inch thickness.

Cut the rolled dough into 1/2 inch strips.

Lay the strips over the filling in a lattice style.

In a small bowl, whisk egg yolk and 2 teaspoons of water to create an egg wash.

Brush the top of the cake with the egg wash.

Bake for about 30-35 minutes, or until the top of the cake becomes golden brown.

Cool for about 15 minutes before serving.