Combine flour, salt, sugar, yeast, and cardamom in the bowl of a stand mixer. Add the 2 Tbsp reserved butter and cut/smash it into the flour mixture like when you make biscuits.
Put the bowl on the mixer and add the water, vanilla, milk, and eggs. use your Thermapen ONE to verify that the water is room temp. Mix using the dough hook until everything just comes together and a sticky, shaggy dough is formed with no dry spots. Turn the dough out onto a floured spot don't encounter to rest.
Make the butter sheets by cutting the butter sticks—between 55 and 63°F (13 and 17°C)—into quarters and lining half of them up side by side on a sheet of parchment. Flour them a little and top with another sheet. Roll them out to a 6" x 9" rectangle. Repeat with the remaining butter to make a second sheet. Set aside.
Flour your work surface well and roll out the dough into a 12" x 24" rectangle with the short edge parallel to the edge of your work surface.
Lay one butter on the center of the dough and fold the close third over it. Lay the second butter atop that fold and fold the remaining dough up onto it. Crimp the left and right sides closed.
Turn the dough packet 90° and roll it out to 24" x "10" and fold the far this to the middle and the near third up onto that. Wrap well in plastic wrap and refrigerate. Set a timer for 20 minutes.
Remove the dough from the fridge and verify that it is between 55 and 63°F (13 and 17°C). Set the short edge of the packet (the one of the ones that is more "open") parallel with your work surface and roll out to 24" x" 20" again. Fold into thirds again. Refrigerate again.
Perform this turning, rolling, and folding operation 2 more times, checking your dough/butter temp each time.
If you like, you can let the dough rest in the refrigerator overnight at this point.
Cut the dough mass into thirds along the direction of the last fold.
Roll each third out in turn to an 8" x 16" rectangle, cutting it into 8 pieces.
Roll out each piece to a 4" x 4" square, fill, if desired, with frangipane, and bring opposite corners together into the center of the square, pressing to adhere. Place each pastry on a baking sheet, 8 to a sheet and cover with plastic wrap to let rise. Repeat with the other two thirds of the dough.
Let rise for 45–90 minutes.
Preheat an oven that has been calibrated using Square DOT to 400°F (204°C). If you ant your pastries shiny, baste each with some egg wash before baking
Bake each tray in turn for 18–22 minuted until beautiful golden-brown and delicious smelling.
Top with pastry cream, if desired, or just eat plain or spread with jam.