Center a rack in the oven and preheat to 375 (F). Prepare a baking sheet by lining with parchment paper or a Silpat mat.
In the bowl of a freestanding electric mixer fitted with a paddle attachment, cream together 1/4 cup unsalted butter and 1/2 cup granulated sugar on medium -high speed until light and fluffy and doubled in size. Add 1 large egg and 1 teaspoon vanilla extract and continue beating until well combined.
Lower the mixer to its lowest setting and add 1 1/2 cups cornstarch, mixing together until just combines. With the mixer still on its slowest speed, gradually add 1/2 cup all-purpose flour and 1 teaspoon baking powder until just combined.
Stop the mixer and knead the mixture with your hands until it starts to form a solid mass and come together. Let stand for 5 minutes.
After 5 minutes, use a 1 tablespoon cookie dough scoop to portion out generous 1 tablespoon portions from the dough. Roll the portions into balls between your palms. The balls should be about the size of a walnut. Place the balls on the prepared baking sheet.
Place 1/4 cup natural unsweetened cocoa powder in one bowl, and 1 tablespoon water in another. Dip the mouth of the bottle into the water, before dipping it into the cocoa powder.
With a dough ball cupped in the palm of your hand, press the mouth of the bottle into the top of the ball to create the mushroom "stem". Clean the mouth of the bottle every 2 to 3 uses with a paper towel.
Return the sculpted mushroom cookies to the baking sheet and bake in the preheated oven for 25 minutes, before turning out onto a wire rack to cool completely.