01 - In a saucepan over medium heat, melt the butter. Continue cooking, swirling often, until the butter turns golden brown and develops a nutty aroma, approximately 5–7 minutes. Transfer the browned butter to a heatproof bowl and refrigerate for 30 minutes, or until just solidified but still pliable.
02 - In a medium bowl, combine the all-purpose flour, cream of tartar, baking soda, and salt. In a large bowl, cream together the browned butter and granulated sugar until light and fluffy. Incorporate the eggs one at a time, followed by the vanilla extract. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Cover the dough and chill for at least 15 minutes.
03 - Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper. On a lightly floured surface, roll out the chilled dough to a 1/4-inch thickness. Use a Christmas tree-shaped cookie cutter to create the cookie shapes. Gather and re-roll any dough scraps as necessary.
04 - In a small bowl, combine the granulated sugar and ground cinnamon for the coating. Dip each shaped cookie into the cinnamon-sugar mixture, ensuring an even coverage.
05 - Arrange the coated cookies on the prepared baking sheets, spacing them approximately 2 inches apart. Bake for 9–11 minutes, or until the edges are just set. Allow the cookies to cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.
06 - To prepare the icing, combine the powdered sugar with milk and a few drops of green food coloring until a thick consistency is achieved. Transfer the icing to a piping bag or use a spoon to apply it to the cooled cookies. Adorn with sprinkles or edible pearls to resemble ornaments.