Preheat your oven to 325°F. Line rimmed baking sheets with unbleached parchment paper and set aside.
Make the cookie dough
In a large bowl, place the flour, xanthan gum, baking powder, salt, confectioners’ sugar and granulated sugar, and whisk to combine well. Create a well in the center of the dry ingredients and add the butter, shortening, eggs and vanilla, mixing to combine after each addition. The dough should come together and will be soft.
Shape the snowmen
Divide the dough into approximately 48 pieces, each weighing a bit less than 1 ounce. Divide each piece of dough in half, and roll one half into a ball. Divide the other half of dough into two more pieces, one about 2/3 and one about 1/3 of the remaining dough, and roll each into a ball.
Place the balls of dough, touching one another, in a stack on the prepared baking sheet. Press down flat on the balls of dough slightly to force them closer together. Repeat with the remaining cookie dough, placing the stacks about 1 1/2-inches apart from one another.
Poke the ends of two chocolate sprinkles in the smallest ball in each stack and press into the dough, for eyes. Place the baking sheets in the freezer for a few minutes until the cookie dough is firm.
Bake the snowmen and decorate
Place the shaped and chilled cookies in the center of the preheated oven, one baking sheet at a time, and bake until the dough is set and the cookies are very lightly brown around the edges (about 12 minutes).
Remove from the oven and, working quickly, press one miniature M&M in the center of the middle of the snowman and one in the center of the bottom, as buttons. Allow the cookies to cool completely on the baking sheets.
Make the glaze and finish the cookies
In a small bowl, place the confectioners’ sugar, and 1 1/2 teaspoons of milk. Mix well, until a thick paste forms. Add more milk by drop, mixing to combine well, until the glaze falls off the spoon very, very slowly. It should be opaque white in color.
Place the glaze in a piping bag fitted with a petal tip or small open piping tip. Pipe a scarf between the top and middle balls of the cooled snowman cookies. Allow to sit until the glaze has set.