To make the frangipane, in a food processor, process the almond flour, sugar, salt, butter, eggs, orange zest, rum, and almond extract together until very smooth, about 30 seconds. If you don’t have a food processor, you can mix it by hand, but it will take much longer and the final consistency will be a little coarser.
Line a rimmed baking sheet with a silicone baking mat or sheet of parchment paper. Place the puff pastry circles on the prepared baking sheet. Brush the outer edge of 1 puff pastry circle with your beaten egg. Spoon the frangipane into the center of 1 circle. Evenly spread out, leaving a 1-inch border free and clear of any frangipane. If you are putting in a fève do so now. Make a little secret mark on the galette to be sure your little one gets the feve (and so no one breaks a tooth). Place the second circle of puff pastry over the top and press down to seal the edges.
Put two fingers on the 1-inch border facing out from the center of the galette and draw the backside of a butter knife towards you. This simple technique will create a visually stunning border that will elevate your pastry-making to an art form. Next, place the galette into your refrigerator for 1 hour.
Preheat the oven to 400°F (200°C).
Cut a pattern into the top of the galette using the backside of a paring knife. Classically, it is semi-circles spiralling out of the center. This is solely for show and does not affect the final flavour so you can skip it. Be sure to score the top and not cut through the dough. Brush the top with the beaten egg, then bake until golden brown, about 30 minutes.
Serve warm or at room temperature.