Line the bottom of an 11 inch spring form cake pan with parchment paper and butter the sides. Set aside. *
Batter
Combine flour and baking powder in a mixing bowl and whisk until well-combined. Set aside.
Using a stand mixer (or a handheld mixer), beat butter until fluffy. Keep beating and slowly add sugar, egg yolks, and water
Add the flour and baking powder mixture into the butter and eggs a tablespoon at a time, beating on a lower speed. Beat until just combined.
Add half of the batter onto an oiled (or lined) spring form pan and spread out evenly into a thin layer. Bake for 20 minutes or until golden brown. Remove from pan and cool on a wire rack.
Bake the rest of the batter the same way using the pan (or use two spring form pans if available).
NOTE: Once you remove the cake layer from the spring form pan and it's cooled a bit, gently flip it over using a big dinner plate and loosen the parchment, but don't throw it away. Flip the layer back onto the parchment paper. You will still bake the meringue on the top of the layer, but if you don't loosen the paper now, it will be hard to get it off later without breaking the meringue.
Meringue
Beat the egg whites until stiff peaks form. Slowly add the sugar to the eggs and beat until dissolved.
Divide the meringue in two and spread evenly over the two baked sheets of cake. Sprinkle with sliced almonds, and bake in the oven for another 15 minutes or until the meringue begins to brown.
Remove from the oven and allow to cool.
Filling
Strain the peaches using a sieve (to get rid of the juice as much as possible) and slice them into eighths.
Beat the whipping cream with sugar and whipping cream stabilizer (if using).
Arrange the peaches over one of the cake sheets with meringue and then evenly cover with all the whipped cream.
Cover with the second cake layer and refrigerate until ready to serve.
Sprinkle with some powdered sugar before serving.
Notes
* 11 inch pans are most available here in Germany. Use 9 inch if you have that. However, you might need to increase the baking time a bit if you use the 9 inch pan so the cake bakes through. Insert a toothpick in the center to make sure it comes out clean.