Step 1 Brew 2 cups of chamomile tea. Put the prunes in a small heat-proof bowl, pour the tea over, and let them soak for about 2 hours, until soft.
Step 2 Drain the prunes, dry them with paper towels, and place them in a small bowl. In a small saucepan, combine one-half cup of the sugar with one-fourth cup water. Bring to a boil over high heat, stirring occasionally to dissolve the sugar. Boil undisturbed for 2 minutes, then cool slightly. Pour the syrup over the prunes, then add two-thirds cup Armagnac to cover. Stir to combine, and push the prunes down so they're submerged. (Add a little more Armagnac, if necessary.) Let the prunes soak, unrefrigerated, overnight.
Step 3Place the apples in a large saucepan, add two-thirds cup sugar and the orange zest. Use the tip of a small knife to scrape the seeds of the vanilla bean onto the apples, then drop in the pod. Stir carefully to combine. Cover and cook over low heat until the apples are soft (but not mushy), about 20 to 25 minutes. Let cool.
Step 4Drain the prunes, reserving the liquid. Cut them into quarters, place in a bowl and pour one-fourth cup of the liquid over them.
Step 5Mix 2 tablespoons of the melted butter with 1 tablespoon sugar, 1 tablespoon of the remaining Armagnac-prune syrup and 2 teaspoons Armagnac. Set aside.
Step 6Two to 3 hours before serving, place a baking sheet on the lowest oven rack and heat the oven to 400 degrees. Lightly brush a 13- to 15-inch round pan (such as a pizza pan) with 1 tablespoon of melted butter.
Step 7Unroll the sheets of filo in front of you and cover with a damp towel. Working quickly so the pastry doesn't dry out, brush the top leaf lightly with melted butter. Fold it in half lengthwise, and brush each side lightly with butter. Place one short end of the folded sheet in the center of the pan, letting the sheet hang over the side of the pan. Repeat with the remaining leaves, arranging them so that they look like the spokes of a wheel (with the inner ends stacked in a hub and the outer ends barely touching).
Step 8Very lightly sprinkle some of the butter-Armagnac mixture over the dough that extends over the edge of the pan. Place the prunes in a 6 1/2 -inch circle in the center of the pastry. Drain the apples, remove the vanilla pod and orange zest and place the apples on top of the prunes.
Step 9To enclose the filling, start with the last sheet placed on the pan. Lift up the outer end and bring it toward the center, twisting the piece once so that the underside faces you. Roll up the end of the strip loosely to form a cup-shaped "rose" and set it flat on the filling, pressing lightly so it adheres. Repeat with the remaining leaves, placing the flowers close together to cover the top of the cake. (Don't worry if a little filling shows through.) Sprinkle the top very lightly with the remaining butter-Armagnac mixture, drizzle with the remaining butter and dust with 1 tablespoon sugar.