Preheat the oven to 400 degrees F.
Place a sheet of parchment paper on your work surface and lightly flour it. Flour your rolling pin as well.
Lay one sheet of puff pastry on the parchment paper and roll the dough out to 10x10 inches or 11 x 11 inches.
Place the item you are using as a template (see Tips & Tricks) and cut out your circle using a pastry cutter, pizza cutter, or a sharp knife. Set aside.
Cut out a second circle. You can score a design into this circle using the tool you used to cut the circle. Do not cut all the way through. Do make 4 to 5 slits to let the steam escape. You can incorporate the slits into the design.
Lift the parchment paper with the dough with the design and put it on a rimmed baking sheet.
Put the prepared puff pastry into the fridge while you make the frangipane/almond cream filling.
Place a fine mesh sieve over a medium bowl and add the almond flour.
Gently push the almond flour through the sieve.
Add the all purpose flour to the almond flour and gently mix together. Set aside.
Cut the butter into tablespoon size pieces (does not have to be exact) and add it to the bowl of a stand mixer fitted with a paddle attachment. You can also use an electric mixer or mix by hand.
Add the sugar to the butter and cream together, 1 to 2 minutes.
Add one of the large eggs and the almond extract to the butter and sugar mixture and beat until combined.
Gently add the dry ingredients to the butter mixture and mix on low speed until completely combined. Make sure you scrape down the sides and up from the bottom so that everything is evenly mixed.
Prepare an egg wash by combining a beaten egg and 1 tbsp of water in a small cup.
Remove the pastry from the fridge.
Spread the almond mixture on the bottom disc of dough (the one without the design).
Smooth out the almond cream evenly (i.e., there should not be a big mound in the center), using an offset spatula. Leave a 3/4 inch border around all of the edges.
Add the egg wash to the border using a pastry brush.
You can add the fève to the almond cream at this point. (See Tips & Tricks for fève ideas.)
Place the other circle of puff pastry on top and press the edges down firmly all around.
You can crimp the edges like a pie at this point. Place your thumb and index finger in a V-shape over the inside of the prepared galette. Push into the space between your fingers using the index finger of the other hand.
Brush the top of the pastry with the rest of the egg wash.
Bake until golden brown, about 40 minutes.