Step 1Melt 4 tablespoons (one-half stick) of butter in a large skillet over medium heat. Place the apple slices in a bowl. Cut the vanilla bean lengthwise in half and, using the tip of a small knife, scrape the seeds over the apples and drop the pod on top. When the butter is foamy, add the apples with the vanilla and the sugar and cook, stirring very gently but frequently, until the apples are lightly caramelized and soft, about 15 to 20 minutes. Add the Calvados and, using a long match and standing well back, set it aflame. When the flames subside, turn the apples over in the Calvados; when the flames have died out and the Calvados has reduced to a glaze, transfer the apples to a bowl and allow them to cool to room temperature.
Step 2Center a rack in the oven and heat it to 350 degrees. Place a 10-inch tart ring on a baking sheet lined with a silicon mat or parchment. Melt the remaining 6 tablespoons butter and set it aside. Unfold the filo dough on your work surface and cover it with a damp towel.
Step 3Remove the top sheet of filo (re-cover the remaining sheets), brush it lightly with butter, and dust it with powdered sugar shaken from a fine-mesh strainer. Gently and loosely crumple the dough into a circle and lay it into the pastry ring. Sprinkle it with about one-fifth of the almonds. Repeat this procedure three more times, until you have four buttered, sugared and almond-sprinkled sheets of filo layered in the ring. Do not press them together -- let them keep some height.
Step 4Spoon the apples into the center of the croustade, leaving a 1-inch border bare. Working as you did before, butter, sugar and crumple a sheet of filo, fitting it over the apples. Sprinkle this layer with the remaining almonds, and cover this with another crumpled sheet of buttered and sugared filo. Do a little styling and draping; arrange the filo so it looks good.
Step 5Slide the croustade into the oven and bake for about 10 to 12 minutes, watching the top of the tart carefully to make certain it doesn't brown too much. The top should be just lightly browned. Remove the croustade from the oven.
Step 6Increase the oven temperature to 400 degrees. Butter and sugar another sheet of filo, loosely crumple it and place it on the last layer to make a light, airy crown. Bake the tart for 5 to 10 minutes, or until lightly browned, then remove it from the oven again.
Step 7Butter the last sheet of filo and, once again, crumple it to make a crown. Place it on top of the croustade and dust it heavily with the remaining powdered sugar. Return the tart to the oven and bake until the top layer caramelizes evenly, about 5 to 10 minutes. Check the progress of the sugar frequently because it can go from brown to burned in a flash. Pull the croustade from the oven as soon as the top is a golden caramel color and allow it to cool for 5 to 10 minutes.
Step 8To serve, lift off the tart ring and, using two large, wide metal spatulas, transfer the croustade to a serving plate. Serve the tart warm or at room temperature the day it is made, with creme fraiche, whipped cream (or even better, creme fraiche lightened with whipped cream) or vanilla ice cream.