Make the Quick Puff Pastry
This quick version of puff pastry produces great butter flavor and plenty of flaky layers. It won’t get you the height of regular puff pastry, but sometimes you don’t want the height.
Put flours and salt in the bowl of your food processor and pulse a few times to mix. Add the butter all at once and pulse about 15 times until the butter is the size of baby peas or smaller. Dump the flour-butter mix into a very large mixing bowl and add the water. With one hand, use a plastic scraper to mix the dough. Do this gently but thoroughly. If you need more water, add it 1 tablespoon at a time until your dough comes together and is not crumbly. Pat into a neat square and refrigerate for 20 minutes.
Meanwhile, make a nice clean space on your counter and sprinkle with flour. Roll the dough out in a 12- x 18-inch rectangle about 2 inches thick. Fold the rectangle in thirds like a letter, then turn 90° and roll again to get a similar-size rectangle. Repeat this 3 times. If your kitchen gets too warm and the dough seems like it’s getting greasy, chill the dough in the fridge for 20 minutes before continuing. After you’ve completed the 4 turns, refrigerate the dough for at least 1 hour.
Make Frangipane, Egg Wash and Assemble
Divide puff pastry in half and refrigerate one half while working with the other. Roll dough to approximately 1/8-inch thickness on a lightly floured surface and cut a 9-inch circle using a cake pan or plate as a template. Brush off excess flour and transfer to a lined cookie sheet. Do the same with the second piece of puff pastry and refrigerate while making the frangipane.
Grind the almonds and sugar in the bowl of a food processor until evenly and finely ground. Add the butter in chunks and pulse until incorporated. Add the egg white and almond extract and pulse until incorporated.
Make the egg wash by thoroughly mixing the yolk, water, and salt in a small cup.
Spread the frangipane filling over the first pastry round to within 1 inch of the edges and lightly brush the border with egg wash, trying not to brush the edge of the pastry itself. Top with the second pastry round. Using a sharp knife or razor, etch a decorative star pattern on top, being careful to not cut through the entire layer. Crimp the edges of the pastry in a scallop pattern (I use the back of a spoon). Brush with egg wash, scatter the sliced almonds on top, and return the pastry to the refrigerator for a half hour while preheating the oven to 400°F. Bake for 10 minutes, then reduce the temperature to 350°F for 25–30 minutes until the pastry is golden.
If you like, you can brush the pastry with a hot simple syrup: Simmer 1 cup water and 1 cup sugar for 1 minute. Add vanilla, brandy, rum, or any liqueur of choice, and brush the pastry while it is still hot. Let cool at least 1/2 hour before serving.