To make the cake: Preheat the oven to 275 degrees F.
Line 2 baking sheets with parchment and draw on 2 (9-inch) circles. Cook's Note: You could also do rectangles.
In a small bowl, combine the ground almonds, cornstarch, and all but 2 tablespoons of the sugar. In a separate bowl, beat the whites to soft peaks, and then beat in the remaining sugar to make a stiff meringue. Fold in the almond mixture in 3 batches to combine thoroughly. Using a piping bag with a large tip, pipe the meringue onto the circles on the parchment. Bake until crisp and dry, about 1 1/2 hours.
To make the buttercream: Heat the sugar with 1/3 cup plus 2 tablespoons water to dissolve, then boil to soft-ball stage, which is just before the syrup changes color (235 to 240 degrees F on a candy thermometer). Beat the egg yolks in a stand-up mixer, then, with the beaters running, quickly add the hot syrup in a thin stream. Continue beating until the mixture is cool and forms and thick mousse, about 5 minutes. Beat in the butter, gradually, and finally add the vanilla.
To assemble the cake: Set a round of meringue on a serving plate. Spread over a quarter of the buttercream. Lay in a layer of sliced peaches. Spread the top layer with buttercream and set it on top of the first. Spread the remaining butter cream over the sides of the cake. Press the toasted almonds all around the sides. Arrange another layer of peach slices on top.