In a small bowl, stir together the yeast, warm milk, and 1 teaspoon of the sugar. Let sit for 10 minutes, until the mixture is foamy and active. Transfer to a stand mixer bowl. Add the remaining sugar, butter, egg, flour, and salt. Mix on medium speed for 10 minutes, until the dough is smooth and elastic. Lightly oil a clean bowl, place the dough inside, grease the top of the dough, and cover. Let rise in a warm place for 1½ hours, or until doubled in size.
Punch down the dough and divide it into 6 equal portions, about 100 grams each. Flatten each piece and place a heaping tablespoon of cookie butter in the center. Fold the dough over the filling, pinching to seal, and roll into a smooth ball. Arrange on a parchment-lined baking sheet. Cover loosely and let rise for another 30 minutes.
While the dough is rising, make the syrup. In a small saucepan, bring the sugar and water to a boil. Remove from heat and let cool completely.
Preheat the oven to 350°F (180°C). Brush the buns with the egg yolk and milk mixture. Bake for 20 minutes, or until deeply golden brown. While still hot, brush generously with the cooled sugar syrup. Let cool completely.
For the topping, whip the heavy cream and powdered sugar until soft peaks form. Fold in the cookie butter gently to create a smooth, creamy filling. Spread or pipe the cookie cream over the cooled buns. Finish with a generous sprinkle of cookie crumbs before serving.