1. Prepare the Puff Pastry.

2. Make the frangipane: In the bowl of a food processor, blend the almonds and the sugar until very fine. Add the butter, egg, rum, flour, and almond extract and process until smooth. Wrap in plastic wrap and shape into a 6-inch round. Freeze until firm, at least 45 minutes.

3. On a lightly floured work surface, roll out puff pastry into a rectangle about 18 by 9 inches and 1/8 inch thick. Using a 9-inch cake pan as a guide, cut two 9-inch rounds from the dough with a very sharp paring knife or pastry wheel. Using a 1-inch circular biscuit cutter or a large round pastry tip, cut out a steam vent from the center of the top round. Place rounds on a baking sheet and freeze until very firm but still pliable, about 20 minutes.

4. In a small bowl, whisk together the egg yolk and heavy cream for the egg wash.

5. Preheat the oven to 375 degrees. Remove dough from freezer. Place frozen frangipane round in the center of the bottom dough circle. Brush the border with egg wash, taking special care not to let the egg wash drip down the sides, which would inhibit proper puffing during baking. Place the second round on top, and press to seal, using your fingers. Using a small paring knife, score the top of each Pithivier in a circular, decorative cross-hatch pattern. With a small paring knife, score around edges in 1/4-inch increments. Transfer Pithiviers to a baking sheet sprayed well with water and chill for at least 1 hour.

6. Remove Pithiviers from freezer. Brush top with egg wash, again being careful not to let any excess drip down over cut edge of dough. Bake until puffed and golden brown, about 45 to 50 minutes.

7. Transfer Pithiviers to a wire rack, and let cool at least 20 minutes before serving.