For the dough:
Combine yeast and milk in the bowl of a mixer fitted with a paddle attachment and mix on low speed.
Slowly add sugar, orange zest, cardamom, vanilla extract, vanilla seeds, eggs, and orange juice. Mix well.
Change the paddle to a dough hook and add salt and flour, one cup at a time, increasing the speed, until the flour is incorporated.
Knead the dough for about 5 minutes, or until it becomes smooth.
Transfer the dough to a lightly floured surface, covering it with plastic wrap.
Refrigerate the dough for 30 minutes.

For the folding:
Roll out the dough on a floured surface to a large rectangle, 18″ x 13″ x 1/4″ (35 cm x 30.5 cm x 0.5 cm).
Spread the butter over 2/3 of the dough and fold into thirds (the unbuttered part in the middle, then the last 1/3 on top).
Refrigerate for 30 minutes.
Turn the dough a quarter turn and roll out again.
Fold into thirds and refrigerate again for 30 minutes.
Repeat this process three times.
After the final turn, refrigerate the dough for 5 hours or overnight.

For the filling:
Mix all ingredients, except butter, in a bowl.
Melt the butter in a saucepan and pour in the mixture.
Let simmer for 6 to 8 minutes, or until the mixture has turned a nice caramel color and has thickened.
Set aside and let cool.

Assembly:
Divide the dough into two equal parts.
Then, roll out the dough and cut a large rectangle to place directly on a baking sheet (covered with a silpat).
Cut the dough using this method (click).
Spread the apple filling, then close the braid.
Let rise for 2 hours in a temperate place.
Brush the braid with the beaten egg.

Preheat the oven to 400 F (200 C).
Bake the braid for 10 minutes, then lower the oven to 350 F (180 C) and continue baking for 15 to 20 minutes.
Let cool on a rack and serve the braid warm or at room temperature.