How to Make It
In a large bowl, whisk the flour with the sugar and salt. Using a pastry blender, cut in the butter until the mixture resembles coarse meal. In a small bowl, whisk the egg with the egg yolk, then whisk in the water. Drizzle the egg over the flour mixture, and stir with a fork to lightly blend the ingredients. Squeeze gently until a smooth dough forms. Divide it in half, pat each half into a disk and wrap in plastic. Refrigerate until firm, about 1 hour.
Preheat the oven to 350°. On a floured work surface, roll 1 of the pastry disks into a rough 9-by-24-inch rectangle about 1/8 inch thick. Using a plate as a guide, cut out three 8-inch rounds. Transfer them to a large baking sheet with plastic wrap between them to prevent sticking. Repeat with the remaining pastry disk. Cut out leaf decorations from the scraps to use for garnish. Refrigerate the rounds and decorations until firm, at least 20 minutes. Remove from the refrigerator and let stand until pliable, about 2 minutes.
Lightly oil a large rimmed baking sheet. Drape a pastry round over each apple and wrap the apples without stretching the pastry, tucking it around and underneath to form a neat package. Transfer the apples to the baking sheet. Using a small knife, make a hole in the pastry at the top of each apple and press the cinnamon sticks on the diagonal through the holes and into the apples, creating stems. Brush the pastry with the beaten egg and bake the apples for about 1 hour, or until they are cooked through and the pastry is nicely browned.
In a large skillet, sprinkle the plums with the sugar and bring to a simmer over moderately high heat. Stir well, cover and simmer over moderate heat, stirring occasionally, until just tender, about 8 minutes. Using a slotted spoon, transfer the plums to a large heatproof bowl. Add the Armagnac to the skillet and simmer over moderately high heat until the cooking liquid is syrupy, about 3 minutes. Add the syrup to the plums.
Let the apples cool on the baking sheet for about 5 minutes. Using a metal spatula, transfer the apples to plates. Serve with the plum sauce.