Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a medium bowl, combine the softened butter, sugar, vanilla, and almond extract. Mix with a spatula until smooth and well combined (do not whisk—avoid incorporating too much air).
Add one whole egg and one egg white (reserve the second yolk for egg wash). Mix until incorporated. Add the almond flour and stir until you have a smooth, thick but spreadable paste. Transfer to a piping bag (no tip needed).
Unfold the puff pastry, roll if needed to a 10 x 15 inch rectangle. Cut into 6 squares, about 5 inches each side. Beat the reserved egg yolk with 1 tablespoon of water. Brush the edges of each pastry square with this egg wash.
Pipe the almond frangipane filling diagonally across the center of each square. Fold two opposite corners toward the center, overlapping slightly, so that a little filling still shows at the tips. Brush the tops with more egg wash, then sprinkle with almond slices.
Place pastries on the prepared baking sheet and bake for 18–22 minutes, or until golden brown and puffed. Check after 15 minutes—move the baking sheet down if browning too quickly, or up if too pale. Cool for 5 minutes, dust with powdered sugar, and serve warm.
