For the puff pastry:

1. Beurre manié: Rub butter with your finger tips, then add the flour in one addition. Rub in together butter and flour until a smooth dough. Shape into a thick, compact square. Wrap into cling film and place in the refrigerator.
2. Puff pastry: mix the flour with the salt, then turn out onto your work surface in a pile. Run your fingers down the center to create a trough. Sprinkle 1 Tablespoon of water into the trough. Quickly fluff the dough with your fingers, keeping your fingers loose and using a scooping motion. Gather the flour back in to a mound, create a trough, and add another Tablespoon of water. Continue sprinkling and fluffing until the flour clumps together in large pieces and holds together when pressed.

Folding the puff pastry:

1. Shape the dough into a cross with 4 arms, keeping the center part a little bit fluffy. Roll out each arm toward the outside, using a rolling pin. Place the beurre manié in the center, then fold each arm of the dough over the butter so they meet in the middle. With the last arm, make sure to cover all over, including the sides.
2. For the 1st turn: using a rolling pin, gently shape into a long rectangle 12 inches by 6 inches (30,5 x 15 cm), tapping the dough rather than rolling it, so that you spread the butter from inside without piercing the dough. Fold the top third over the bottom third, and the bottom over the top third, like a letter. The first tour is done.
3. For the other turns: rotate the folded dough so it looks like a book about to be opened. Roll it out again into a rectangle 12 inches by 6 inches. Fold it again. Make 6 turns in total. My advice: start with 3 tours, then wrap the dough into cling film and place in the refrigerator. Remove a few hours later, ideally the day after, and make the 3 remaining turns.

For the frangipane:

1. Crème pâtissière* : heat milk with vanilla in a medium saucepan, and stop when it starts bubbling. In a medium-size bowl, beat yolks with sugar until pale and fluffy. Add in the flour and hot milk, and whisk vigorously. Put it back in the saucepan and cook for a few minutes until the mixture thickens, whisking constantly. Pour into a medium-size bowl, and cover with cling film, creating contact with it to prevent from drying on top. Let chill in the refrigerator.
2. Almond cream (crème d’amande): in a medium-sized bowl, stir in butter and sugar until creamy. Add in the eggs, one at a time, whisking between each addition. Pour the rum and the almond flour, and stir well. Grate the tonka bean and add in to the preparation (you can replace the tonka bean with vanilla extract).
3. Withdraw 3.5 oz (100g) of crème pâtissière and combine with almond cream. You get a so-called crème frangipane*. Transfer the frangipane into a piping bag and place in the refrigerator until ready to use.

To assemble the galette des rois:

1. Roll out the puff pastry in a long stripe. Cut in two circles, one of about 10.5 inch (28 cm), and the over one about 0.3 inch (1 cm) smaller.
2. Place the smaller disk onto a prepared baking sheet lined with parchment paper, then pipe the frangipane into circles, starting from the center, leaving about 0.5 inch (1,5 cm) margin all over. At this point, don’t forget to hide the fève. Cover with the second disk, and press with your thumb and make small cuts with a knife all around.
3. Whisk the yolk with maple syrup. Brush the galette des rois, and draw some patterns of your choice on top, using a small knife (just draw, do not cut through!).
4. Bake for about 30-40 minutes at 350°F (180°C), or until the galette is golden-brown.