If making your own pastry, follow my recipe to make two sheets of Rough Puff Pastry. If using store-bought puff pastry, follow package instructions to thaw.
Roll the rough puff pastry out until it’s between ⅛” thick and cut out two 8” or 9” circles.
To do this, lightly four the rim of the appropriate sized cake pan or bowl and place on the dough. You can also cut out a circle from paper to use as a stencil. Using a sharp paring knife, cut out the pastry circle by running it the whole way around the edge of the bowl. Repeat the process again with the remaining dough so there are two circles.
Take one circle and gently roll again two times one way, and two times the other way to make this circle slightly larger. This will be the top circle, and helps to make sure the pastry edges meet once placed on top of the filling.
Place both circles on a sheet pan lined with parchment paper and set in the refrigerator while you make the almond filling.
In a stand mixer fitted with the paddle attachment (or with a hand mixer), beat the butter and powdered sugar together for about one minute. Add the almond flour, all purpose flour, and salt and mix on low until combined. Add the almond extract, vanilla, and one egg and mix on medium low until the egg is fully incorporated. You can also do this by hand with a wooden spoon, just try to combine it as thoroughly as possible (use a little elbow grease!).
Remove the dough from the refrigerator and pipe or spread the almond filling on the smaller circle (the bottom), keeping a 1” border. If piping, pipe a full coil inside the border, then a smaller diameter coil on top. If spreading, keep the almond filling mounded in the middle.
Brush a small amount of water around the filling the whole way to the edges (but not over). Add the top piece and firmly press down along the border to seal the pieces of rough puff together. Press the whole way in to the filling, then gently flatten out the top by smoothing your hand overtop.
Using the back of a butter knife (or the mouth of a spoon) with the tip touching the pan, gently press the knife against the edge towards the center to create a small indent. Continue this the whole way around the dough, 1” apart.
Separate one egg and whisk the yolk together with a small splash of water. Brush the yolk all over the top and border of the pastry, careful not to get any of the egg wash along the sides (or else the flaky edges will be lost).
Place in the refrigerator for 10 minutes. Add another round of egg wash, and rest in the refrigerator for another 10 minutes.
While the dough is chilling for the second time, preheat the oven to 425°F/218℃.
Remove the dough and lightly score a design into the top piece of dough, taking care not to pierce through the dough. Using the knife or a toothpick, poke about 5-6 holes through the top of the dough to allow steam to escape.
Reduce the oven to 400℉/205℃, then bake for 30-40 minutes until the pastry has risen and become a deep golden brown.
Slice and serve warm or at room temperature.