Preheat the oven to 425 degrees F. Line a 9-inch pie plate with the dough; crimp the edges. Pierce the bottom all over with a fork, then refrigerate 15 minutes. Line the crust with foil and fill with pie weights or dried beans. Place on a baking sheet and bake until set, about 7 minutes. Remove the foil and weights; continue baking until golden, about 6 more minutes. Transfer to a rack to cool.
Meanwhile, spread the almonds on a baking sheet and bake until golden, about 8 minutes. Combine the apples, 5 tablespoons sugar, the butter, lemon juice and vanilla in a saucepan; cover and cook over medium heat until the apples are tender, about 12 minutes. Uncover and cook, mashing and stirring with a potato masher, until saucy and starting to caramelize, about 12 more minutes; let cool 15 minutes. Transfer the apples to the pie crust and sprinkle with the toasted almonds.
Beat the egg whites and salt with a mixer on medium speed until foamy, about 2 minutes. Gradually add the remaining 6 tablespoons sugar and beat until stiff, 2 to 3 more minutes. Transfer the meringue to a pastry bag fitted with a 1/2-inch star tip. Pipe around the edge of the pie, then pipe in a lattice pattern. Bake until the meringue is golden, 15 to 20 minutes. Let cool completely.