Combine the yeast and milk in the bowl of a mixer fitted with the paddle attachment and mix on low speed.
Slowly add sugar, orange zest, cardamom, vanilla extract, vanilla seeds, eggs, and orange juice. Mix well.
Change to the dough hook and add the salt with the flour, 1 cup at a time, increasing speed to medium as the flour is incorporated. Knead the dough for about 5 minutes, or until smooth. You may need to add a little more flour if it is sticky.
Transfer dough to a lightly floured baking sheet and cover with plastic wrap. Refrigerate for 30 minutes.
Without a standing mixer: Combine yeast and milk in a bowl with a hand mixer on low speed or a whisk. Add sugar, orange zest, cardamom, vanilla extract, vanilla seeds, eggs, and orange juice and mix well.
Sift flour and salt on your working surface and make a well. Make sure that the “walls” of your well are thick and even.
Pour the liquid in the middle of the well. With your fingertips, mix the liquid and the flour starting from the middle of the well, slowly working towards the edges. When the ingredients have been incorporated start kneading the dough with the heels of your hands until it becomes smooth and easy to work with, around 5 to 7 minutes. You might need to add more flour if the dough is sticky.
Combine the butter and flour in the bowl of a mixer fitted with a paddle attachment and beat on medium speed for 1 minute. Scrape down the sides of the bowl and the paddle and then beat for 1 minute more, or until smooth and free of lumps. Set aside at room temperature.
After the butter block has chilled for 30 minutes, turn it out onto a lightly floured surface.
Roll the dough into a rectangle approximately 18" x 13" and ¼" thick. The dough may be sticky, so keep dusting it lightly with flour.
Spread the butter evenly over the center and right thirds of the dough.
Fold the left edge of the dough to the right, covering half of the butter. Fold the right third of the rectangle over the center third. The first turn has now been completed.
Mark the dough by poking it with your finger to keep track of your turns, or use a sticky and keep a tally.
Place the dough on a baking sheet, wrap it in plastic wrap, and refrigerate for 30 minutes.
Place the dough lengthwise on a floured work surface. The open ends should be to your right and left. Roll the dough into another approximately 13" x 18", ¼"-thick rectangle.
Once again, fold the left third of the rectangle over the center third and the right third over the center third. No additional butter will be added as it is already in the dough. The second turn has now been completed. Refrigerate the dough for 30 minutes.
Roll out, turn, and refrigerate the dough two more times, for a total of four single turns. Make sure you are keeping track of your turns.
Refrigerate the dough after the final turn for at least 5 hours or overnight. The Danish dough is now ready to be used. If you will not be using the dough within 24 hours, freeze it. To do this, roll the dough out to about 1" in thickness, wrap tightly in plastic wrap, and freeze. Defrost the dough slowly in the refrigerator for easiest handling. Danish dough will keep in the freezer for up to 1 month.
Toss all of the ingredients, except the butter, into a large bowl.
Melt the butter in a sauté pan over medium heat until slightly golden in color, about 6 – 8 minutes.
Then add the apple mixture and sauté until apples are softened and caramelized, 10 to 15 minutes. If you’ve chosen Fujis, the apples will be caramelized, but have still retained their shape.
Pour the cooked apples onto a baking sheet to cool completely before forming the braid. (If making ahead, cool to room temperature, seal, and refrigerate.) They will cool faster when spread in a thin layer over the surface of the sheet. After they have cooled, the filling can be stored in the refrigerator for up to 3 days. Left over filling can be used as an ice cream topping, for muffins, cheesecake, or other pastries.
Line a baking sheet with a silicone mat or parchment paper.
On a lightly floured surface, roll the Danish Dough into a 15" x 20" rectangle, ¼" thick. If the dough seems elastic and shrinks back when rolled, let it rest for a few minutes, then roll again. Place the dough on the baking sheet.
Along one long side of the pastry make parallel, 5" long cuts with a knife or rolling pastry wheel, each about 1 inch apart. Repeat on the opposite side, making sure to line up the cuts with those you’ve already made.
Spoon the filling down the center of the rectangle.
Starting with the top and bottom “flaps”, fold the top flap down over the filling to cover. Next, fold the bottom “flap” up to cover filling. This helps keep the braid neat and helps to hold in the filling.
Now begin folding the cut side strips of dough over the filling, alternating first left, then right, left, right, until finished. Trim any excess dough and tuck in the ends.
Whisk together the whole egg and yolk in a bowl and with a pastry brush, lightly coat the braid.
Proofing and Baking.
Spray non-stick spray onto a piece of plastic wrap and place over the braid. Proof at room temperature or, if possible, in a controlled 90°F area for about 2 hours, or until doubled in volume and light to the touch.
Near the end of proofing, preheat oven to 400°F Position a rack in the center of the oven.
Bake for 10 minutes, then rotate the pan so that the side of the braid previously in the back of the oven is now in the front.
Lower the oven temperature to 350°F, and bake about 15-20 minutes more, or until golden brown.
Cool and serve the braid either still warm from the oven or at room temperature. The cooled braid can be wrapped airtight and stored in the refrigerator for up to 2 days, or freeze for 1 month.