Beat the butter and sugar in a large bowl with a mixer on medium-high speed until light and fluffy, 3 to 5 minutes. Add the egg and vanilla and beat until incorporated. Reduce the mixer speed to low; add the flour mixture in 2 batches and beat until just incorporated. Divide the dough in half, wrap in plastic wrap and refrigerate until firm, at least 1 hour and up to 1 day.
Line 2 baking sheets with parchment paper. Working with 1 piece of dough at a time, dust the dough generously with flour and roll out between 2 pieces of parchment paper until about 1/8 inch thick. Cut out shapes using 2-to-4-inch cookie cutters and transfer to the prepared baking sheets. Gather the scraps and refrigerate until firm; reroll once to cut out more cookies. Refrigerate the cutouts until firm, about 30 minutes.
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Bake the cookies, switching the position of the pans halfway through, until slightly puffed and darker around the edges, 12 to 15 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely. Sandwich the cookies with the marshmallow cream.