Prep. Preheat the oven to 350°F and line baking sheets with parchment paper. Set aside.
Cream the butter and sugar. Using an electric mixer, cream the softened butter and sugar in a large mixing bowl until light and fluffy. Add the egg and vanilla extract, and continue mixing until smooth and well combined.
Combine the ingredients. In a separate bowl, sift the flour, baking powder, and salt. Gradually add the dry ingredients to the wet mixture, mixing on low speed until a soft dough forms.
Roll and cut the cookies. Lightly flour a work surface and roll out the dough to ¼ inch thickness. Use a 2–3 inch round cookie cutter to cut out shapes. Carefully transfer the cutouts to the prepared baking sheets, spacing them about 2 inches apart.
Bake the cookies. Bake the cookies in the preheated oven for 12-14 minutes, or until the edges are lightly golden. Remove the baking sheets from the oven and let the cookies cool for 5 minutes before transferring them to a wire rack to cool completely.
Prepare the icing. In a small bowl, mix the sifted icing sugar with milk, one teaspoon at a time, until a smooth and slightly thick consistency is achieved. Adjust as needed to ensure the icing spreads easily but doesn’t run off the cookies.
Decorate the cookies. Spread the icing over each cooled cookie to resemble melted snow. Place a marshmallow on one side of each cookie (over the icing so it sticks) to act as the snowman’s head. Use edible food colors to draw faces on the marshmallows and add decorative elements like buttons, arms, or scarves directly on the icing.
Set and serve. Allow the decorated cookies to sit for about 30 minutes or until the icing has fully set, and then enjoy these adorable Melted Snowman Cookies!