Cream the softened butter and granulated sugar until light and fluffy. Beat in the egg until smooth. Add almond flour, cornstarch, vanilla, and dark rum; mix until smooth and spreadable, then refrigerate.

Divide thawed puff pastry in half, keeping one piece chilled while rolling one half into an 8-inch circle about 1/8 inch thick. Place on a parchment-lined baking sheet and chill. Cut a slightly larger circle for the top and keep chilled.

Prick the base circle with a fork to limit puffing under the filling. Spread frangipane over the center, leaving a 1-inch border. Brush the border lightly with water, and lay the larger pastry circle over the top, sealing the edges firmly.

Shape a scalloped edge by pressing a butter knife between your fingers all around the pie. Cut a small steam vent in the center and score curved lines. Brush with beaten egg yolk, avoiding the cut edges.

Chill the assembled pithivier for at least 30 minutes while preheating the oven to 375°F. Bake for 35 to 40 minutes until deep golden brown. Cool for at least 15 minutes before slicing.

Store baked leftovers covered in the refrigerator for 1-2 days. For best texture, reheat in a 350°F oven before serving.