Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.

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.

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.

Brush uncovered puff pastry edge with a little water and lay the second puff pastry circle on top. Seal the edges by pressing together.

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.

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.

Remove from the oven and dust with confectioners' sugar.

Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.

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.