In the bowl of a stand mixer fitted with the paddle attachment, beat butter, sugar, vanilla, and salt at medium speed until creamy, 2 to 3 minutes, stopping to scrape sides of bowl.

In a medium bowl, whisk together flour and custard powder. With mixer on low speed, gradually add flour mixture to butter mixture, beating until dough begins to clump together. Turn out dough onto a lightly floured surface, and shape into a disk. Wrap in plastic wrap, and refrigerate until chilled, about 1 hour.

Preheat oven to 350°F (180°C). Line baking sheets with parchment paper.

On a lightly floured surface, roll dough to ⅛-inch thickness. Using a 3-inch fluted round cutter, cut dough, rerolling scraps as necessary, and place 1 inch apart on prepared pans. Using desired cookie stamp* dipped in flour, firmly press dough. (Work quickly so dough does not get warm and sticky.) Freeze until firm, about 10 minutes.

Bake until edges are beginning to brown, 10 to 12 minutes. Let cool on pans for 10 minutes. Remove from pans, and let cool completely on wire racks.

Spoon Vanilla Bean Buttercream into a pastry bag; cut a ¾-inch opening in tip. Pipe 2 tablespoons (26 grams) buttercream onto flat side of half of cookies. Place remaining cookies, flat side down, on top of buttercream.