For the filling, beat the heavy cream, vanilla extract, and sugar until stiff peaks form. Slice the peaches into thin pieces. If your cake layers are cold, it will be much easier to cut them into four layers.
Brush the first cake layer with some peach juice. Pipe a thin layer of whipped cream around the edges, then arrange peach slices in the middle. For the second layer, spread only the dulce de leche. Repeat the peach juice, whipped cream, and peach layer for the third and final cake layers, making sure to reserve enough peach slices for the top.
Spread a thin layer of whipped cream around the sides of the cake. If you like, pipe some whipped cream roses around the edges. Heat the remaining peach juice. In a small bowl, mix the cornstarch with a bit of water until there are no lumps, then add it to the hot peach juice. Stir until the glaze thickens, then pour it over the peaches on the top layer of the cake. Crush the dried meringue and gently pat it around the sides of the cake. Serve immediately.