In a stand mixer with a dough hook, combine flour, yeast, and sugar. Add egg, milk, butter, vanilla, and salt; knead until smooth and slightly sticky, 6–8 minutes. Transfer to a greased bowl, cover, and let rise overnight in the refrigerator or at room temperature until doubled.
Line a 9-by-13-inch/20-by-30-centimeter pan with parchment.
Roll or press the dough to fit. Cover and let rise for 30 minutes.
In a small saucepan, melt butter with sugar and honey until dissolved. Stir in almonds, cool briefly, and spread over dough.
Bake at 340°F/170°C until golden, about 30 minutes. Cool completely.
Split the cake horizontally. Boil sugar and water until dissolved, then stir in almond extract. Brush syrup on cut sides.
Whip cream, milk, pudding mix, sugar, and vanilla paste until stiff. Spread over the bottom layer, then top with the almond-crusted half. Chill at least 1 hour before serving.