Step 1
Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Step 2
Combine butter and sugar in a bowl and beat with an electric mixer until well combined. Add almond flour 2 egg yolks and rum; mix until frangipane is smooth.
Step 3
Roll out puff pastry sheets and cut into 2 equal circles. Place the first circle on the baking sheet and pierce several times with a fork. Spread frangipane on the circle leaving a 1/2 inch border.
Step 4
Brush uncovered puff pastry edge with a little water and lay the second puff pastry circle on top. Seal the edges by pressing together.
Step 5
Using a sharp knife cut a decorative pattern around the rim. Trace decorative lines onto the surface without cutting all the way through. Beat remaining egg yolk with a fork; brush over the top.
Step 6
Bake in the preheated oven for 10 minutes. Reduce oven temperature to 350 degrees F (175 degrees C) and bake until pastry is puffed up and lightly browned about 20 minutes more.
Step 7
Remove from the oven and dust with confectioners' sugar.
Step 8
Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.
Step 9
Place pithiviers under the broiler and broil until a golden shiny crust forms 1 to 2 minutes. Watch carefully to make sure it doesn't burn.