Combine the apples, white sugar, brown sugar, cornstarch, cinnamon, cardamom, and salt together in a bowl; mix until well combined.
Heat the butter in a nonstick skillet over medium-high heat. Add the apple mixture and cook, stirring often, until boiling. Reduce the heat to medium and continue to cook, stirring often, for another 3-4 minutes or until the apples are tender and the sauce has thickened. Remove from the heat and set aside to cool and thicken.
Preheat the oven to 400 degrees. Line a baking sheet with parchment flavor.
Remove a puff pastry sheet out of the freezer and thaw on the parchment-lined baking sheet.
Unfold your defrosted puff pastry on the parchment-lined baking sheet. Use the lines already on the folds to guide you on where to make cuts. Using a sharp knife, cut slanting strips along both sides of the pastry, making sure to keep the center untouched.
Spoon the apples down the center portion of the puff pastry then braid the pastry by crossing the strips over the filling overlapping each other. Side Note: Try not to get too much liquid with the apples or it will leak out during baking.
Brush the top of the puff pastry with the egg wash. Sprinkle the top with coarse sugar and thinly sliced almonds.
Bake for 25-30 minutes, or until golden brown and puffed up. Remove from the oven and cool on a cooling rack.
Make the glaze by combining the powdered sugar together with the milk and vanilla extract; mix until creamy and smooth.
Once the braid has cooled completely, drizzle the top with the glaze, slice, and serve. Enjoy.