Beat the egg with a whisk in a medium bowl, then use a spatula to incorporate the almond flour, granulated sugar, rum, almond extract, and melted butter before refrigerating the mixture. The filling should look evenly moistened, without streaks of egg or pockets of dry almond flour.

Work the melted butter and both sugars together with a spatula in a large bowl, blend in the egg, almond extract, baking soda, and salt, then incorporate the flour. Stop once the dough looks uniform and you can no longer see loose flour along the bowl’s bottom or sides.

Portion the dough with a quarter-cup scoop onto a plate or tray, retrieve the chilled filling, flatten each dough portion into an even disc, enclose a two-tablespoon scoop of filling, roll it smooth, press sliced almonds onto the top, and refrigerate the assembled cookies while the oven preheats. Each finished cookie should remain a round ball with a fully closed seam and no darker almond filling showing through the pale dough.

Heat the oven to 350 °F with the rack centered, arrange 8 chilled dough balls on a parchment-lined half-sheet pan, bake for 14 to 16 minutes, cool on the pan for 5 minutes, transfer to a plate or wire rack, dust with powdered sugar, and repeat with the remaining cookies. At the end of baking, the edges should just begin turning golden while the centers still look pale and puffy.