For the base, pour flour onto a flat work surface and shape into a mound. Mix with salt and sugar and create a well in center. Cut cold butter into small pieces and add to flour. Crack egg into well, add 2 tablespoons cold water and chop all ingredients together with a knife. Knead with hands to form a smooth dough, shape into a ball, wrap in plastic wrap and refrigerate for 30 minutes.
For the filling, peel apples, cut into quarters, remove cores and cut into slices. Saute in butter. Add sugar, lemon juice and cinnamon stick. Simmer while stirring occasionally for about 20 minutes or until apples are soft and falling apart. Remove cinnamon stick, remove from heat and cool. Stir in vanilla pudding packages. Set aside 5 tablespoons of apple mixture.
Preheat oven to 160°C (approximately 320°F).
Press dough by hand into pan, pushing up sides to make a thin edge. Sprinkle with almonds and pour in apple mixture. Smooth.
Beat egg white until stiff and fold in reserved apple mixture. Spread on top of filling and bake in oven for 1 hour. Remove from oven, cool and sprinkle with cinnamon sugar. Serve to taste with whipped cream if desired.