Preheat the oven to 375F
In a stand mixer fitted with a paddle attachment, cream together the butter, sugar, and molasses. Mix in the applesauce.
Shape into a ball, wrap in plastic wrap and chill in the fridge for 15 minutes.
Turn out the dough onto a lightly floured surface. Roll out to 1/4 inch thickness.
Using a glass or a round cookie cutter that is slightly larger than your cookie stamp, cut out circles in the dough. Lightly dust your cookie stamp with flour, then press it into the dough using even pressure.
Transfer the cookies to a parchment-lined baking sheet and bake for 8-10 minutes. Be careful not to over bake the cookies - they will be soft when the come up but will firm up a bit.
Allow to cool for several minutes before transferring to a wire rack.