You should definitely use butter!
For the yeast dough, sift the flour into a large bowl, make a well in the center and add the yeast. Spread the sugar, butter and salt around the edges and pour in the warm mineral water.
Knead everything into a smooth dough and let it rise in a warm place until it has about doubled in volume (about 1 hour).
For the crumble dough, knead butter, sugar, flour and vanilla sugar with your hands and work into coarse crumbles.
Divide the yeast dough into 15 pieces, roll out into equal-sized patties and top with the crumbles. Bake in a preheated oven at 200° degrees for about 15 minutes. After cooling, mix the water with the powdered sugar (must not be too liquid) and spread on the patties.
Variation: You can also put chopped apples, apple pulp or jam on the dough and then put the crumble on top and bake. The crumble can be modified by adding cinnamon, nuts or poppy seeds.