Grease a baking tray or tart tin (30 cm Ø) and dust with flour. Mix 300 g flour and icing sugar. Add the egg and 150 g butter in pieces. Knead first with the dough hooks of the mixer, then briefly with your hands to a smooth dough.
Roll out the dough on a lightly floured work surface until round (approx. 38 cm Ø). Carefully lift into the mould and press the edge lightly. Chill for about 30 minutes.
Peel, quarter, core and cut apples into pieces. Mix with lemon juice. Melt 50 g butter in a pot. Sprinkle in 3 tablespoons of sugar. Add the apples with lemon juice and 450 ml orange juice.
Bring to the boil and simmer covered for about 2 minutes. Stir the rest of the orange juice and pudding powder until smooth. Stir into the apples and simmer for 1 minute, stirring gently. Remove the pot from the heat and let it cool down a little.
Preheat oven (electric cooker: 175 °C/circulating air: 150 °C/gas: see manufacturer). Melt 125 g butter. Mix 150 g flour, 50 g sugar and brown sugar well. Add butter. Knead everything into crumbles.
Spread the apple mixture on the base in the mould, sprinkle with crumbles.
Bake the cake in a hot oven on the lowest shelf for about 50 minutes. Remove from the oven and leave to cool in the mould on a cake rack (preferably overnight). Serve with semi-stiff whipped cream and vanilla sauce.