Preheat the oven to 350°F/ 180°C.
Whisk together the flour and the salt.
Dice the butter into small cubes. Work it into the flour until it’s well-distributed, but not fully incorporated. Larger, pea-sized pieces of butter will be scattered throughout the mixture.
Toss dough with a fork or your fingers, drizzle in the ice water. Stop adding water when the dough starts to clump and then take it in your hands. If it holds together easily, without crumbling, it’s ready. If it has dry spots, or pieces break off easily, add a bit more water until it’s totally cohesive.
Gather the dough into a ball. Gently shape into a rough disk. For better texture, chill the dough for at least 30 minutes before rolling. If dough has been refrigerated longer than 30 minutes, allow it to warm at room temperature for 10 to 15 minutes, until it rolls easily.
Roll the dough out to 10 inches/ 25.4 cm in diameter, transfer the dough to the pan.
Mix ground almonds, sugar, and salt in a mixing bowl.
Add egg whites to dry ingredients in a mixing bowl and mix with a spatula or process all the ingredients in a food processor. The dough should be sticky and smooth.
Spoon almond filling on top of crust and spread evenly.
Whisk the baking powder into the flour and set aside.
Beat together the butter, sugar, and salt, until fluffy and lightened in color.
Add the eggs one at a time, beating well after each addition. Scrape the sides and bottom of the bowl once all the eggs have been added, and beat again.
Add the flour mixture to the batter in three parts alternately with the milk, starting and ending with the flour, stir in vanilla extract and mix until everything is well combined.
Spoon the batter over the almond mixture. Level and smooth the top with a spatula.
Spoon small amount of pie filling with a teaspoon over the batter leaving areas of cake batter uncovered.
Bake the cake for 45 to 60 minutes, or until a toothpick inserted into the center comes out clean.
