For the Topping:
For the Dough: In a large measuring cup or medium bowl, sprinkle the yeast in the water to soften. Heat the cream to 100 degrees F and add it to the yeast along with the egg yolks and egg white. Whisk to combine.
In a large bowl, combine the sugar, salt, cardamom, and flour. Using a pastry blender or 2 knives, cut the butter into the flour mixture until it is the size of tiny peas. This process also can be done easily in a food processor. Add the yeast mixture to the dry ingredients and stir just until combined. Do not mix too much because you do not want to soften the cold butter. You should see small flour-coated pieces of butter throughout the dough.
Overnight rise: Put the dough in an oiled bowl and turn to coat the entire ball of dough with oil. Cover with a piece of plastic wrap and a tightly woven towel and refrigerate 12 hours or overnight.
Shape: Turn the dough out onto a lightly oiled work surface and divide in half. Roll each half into a 24-inch square. Fold the dough in half from top to bottom, then fold the dough in half from side to side to make a 12-inch square. Cover with a towel and let rest 10 minutes.
Working with 1 square of dough at a time, roll the dough into a 12 by 24-inch rectangle. Spread half of filling evenly over two-thirds of the dough, filling an area of 8 by 24 inches. Fold the portion of dough that has no filling onto the center third of filled dough, then fold the other third of dough to the center, forming a 4 by 24-inch rectangle.
Place the dough seam side down on a parchment-lined or well-greased baking sheet. Bring the ends of the dough almost together (about 4 inches apart) to form a horseshoe. Repeat with the second piece of dough. If you cannot get both Kringles on a baking sheet, cover the second one and put it in the refrigerator until the first Kringle finishes baking; then bake.
Second Rise: Cover the with a tightly woven towel and let rise for 1 hour.
Preheat Oven: About 10 minutes before baking, preheat the oven to 375 degrees F.
Final Preparation: For the topping, beat the egg white with a whisk or fork until frothy and brush over the loaves. Sprinkle with sliced almonds and then granulated sugar.
Bake and Cool: Bake for 25 minutes until the internal temperature of the bread reached 190 degrees. Immediately remove the bread from the baking sheet and place on a rack to cool.
In a medium bowl, stir the sugar, apples, pecans, butter, and cinnamon together until well mixed. Use to fill 2 loaves.