Add the water to the saucepan containing the sealed milk.l Add the yeast and stir to dissolve.
In the bowl of an electric mixer using the paddle attachment, combine the flours, sugar, salt and cardamom. Blend the dry ingredients on low speed until they are well combined.
On low speed gradually add the warm yeast and milk mixture, softened butter, and eggs until a soft dough forms. Stop the machine and feel the dough. If it feels too sticky, sprinkle in a small amount of bread flour.
Remove the dough from the bowl and knead it until it feels smooth and elastic but still soft. Do not overwork the dough. Shape the dough into a rectangular disk. Wrap the dough in plastic wrap and chill it for 30 minutes.
Preparing the Butter to be enclosed.
Place the butter on a piece of plastic wrap that is at least two or three times its size o a clean, dry work surface. Cover the butter with another piece of plastic wrap the same size as the first. Using a rolling pin, hit the butter to soften it slightly and then begin to roll and push it into an even 6 x 12 inch(15 x 30 cm) rectangle. Do not over handle the butter. It should, ideally, be the same consistency as the dough. Dry it aside, do not chill.
Enclosing the butter On a lightly floured work surface, roll the chilled dough into a 9 x 18 inch (22.5 x 45 cm) rectangle. Place the rectangle of butter across two thirds of the rectangle and fold like a letter, bring the unbuttered end up over to the middle of the butter, sealing the dough all along the edges. Fold the remaining buttered portion of the dough on top, completely sealing the butter inside. Be sure to stretch the corners of the dough to square it off, pressing it down to seal the edges together. Using a pastry brush, dust off any excess flour.
Completing One Three-fold or Letter-fold turn:
Rotate the dough 90 degrees so that when the dough is rolled out, the open ends will become the short sides of the rectangle. It will be these short sides that are folded into the center to make sure the fat stays enclosed. Roll out the dough to a 9 x 18 inch(22.5 x 45cm) rectangle. Using a pastry brush, dust off any excess flour.
Take one short end of the dough and fold it one third of the way toward the middle, like a letter, pressing it down slightly. Take the other short end and pull it over the dough to meet the other side as if you are folding a letter. Press the dough down slightly so the edges remain sealed. This step completes one letter-fold turn. Press one finger into the dough to show one turn is complete. Wrap the dough in plastic wrap and chill for 1 hour. A marker can also be used on the plastic to mark one completed turn.
Repeat rolling, folding, chilling and marking the number of completed turns two more times for a total of three turns. After the third and final turn is complete, wrap the dough tightly in plastic wrap two times and chill it overnight in the refrigerator. The dough will expand in the refrigerator because of the yeast, so be sure it is swaddled securely within the plastic.
To make the Almond Filling:
In a food processor, process cake pieces until finely ground.
Add the almond paste, sugar, and butter and process the mixture until well combined. With the motor running, add the water and process the mixture until a paste forms.
To assemble the Almond Danish Braid :
Roll the danish dough out to a 10 x 15 inch(25 x 37.5 cm) rectangle on a lightly floured surface. Brush off the excess flour from the dough. Pick up the dough and place it onto a parchment lined sheet pan.
With a pizza cutter, lightly mark a line parallel to the long side approximately 3 inches in from the edge, going completely down the length of the dough without cutting through to the other side. Do the same on the opposite long side.
Spread the almond filling over the narrow rectangle that has formed. Scatter half the almonds over the filling. Using a pizza cutter, make 3 inch(7.5 cm) long cuts on an angle that are approximately 3/4 inch(2 cm) apart all the way down the length of the dough on both sides. These cuts should resemble fringe on both sides of the dough.
Lightly egg wash the fringed strips. Begin at the end of the dough where you started to cut the fringed pieces and fold the piece of the dough on top of the narrow rectangle to come over the filling by 1 inch(2.5 cm). Fold in the strips alternating left and right sides and continuing down the full length of the dough so that it resembles a braid. Reserve the egg wash by covering it with plastic wrap and place it in the refrigerator.
Place the braid in a proof box at no higher than 85f(29c) for 30 minutes to 1 hour or until the braid appears puffy.
Preheat the oven to 375f(190c)
Using a pastry brush, brush the entire surface of the braid with the reserved egg wash.
Bake the braid for 25 - 30 minutes or until the braid is golden brown.