Preheat your oven to 350° and line a half sheet pan with parchment paper or spray with a cooking spray.
In the bowl of your standing mixer, use the paddle attachment to cream the butter and granulated sugar together. Use a medium speed to beat the butter and sugar until they are light and fluffy - about 3 minutes.
While the butter and sugar are mixing, combine the flour, ginger, cinnamon, cloves, baking soda, and salt in a small bowl. Stir to combine.
Stop the mixer and add the egg, molasses, and milk. Mix until well combined.
Stop the mixer and add ½ of the dry ingredients. Turn your mixer on low and combine. Add the second ½ of the dry ingredients. Mix again until the ingredients are just combined.
Using a small (½ tablespoon) cookie scoop, drop 16-20 balls of dough onto your prepared sheet pan. If you don't have a small cookie scoop, use a teaspoon to scoop the dough, and then use your hands to roll the dough into balls. Roll the cookie dough in granulated sugar and place on your cookie sheet.
Bake the dough at 350° for 7-8 minutes. You want the cookies to be firm, yet not overcooked.
Cream Cheese Buttercream
In the bowl of your standing mixer, using the paddle attachment to whip the softened butter and cream cheese until they are light and fluffy - about 5 minutes.
Add the confectioner's sugar, vanilla, and whole milk.
Turn your mixer on low and combine all ingredients. Once combined, turn the mixer to medium high and whip the buttercream until it is light and fluffy, about 2-3 minutes.
Stop the mixer, scrape the sides of the bowl, add a pinch of salt to the buttercream and mix once more, on medium high, for about 1 minute.
Making the Sandwiches
Once the cookies are completely cool, flip ½ of the cookies over so that the flat side is facing up.
Fit a pastry bag with a plain pastry tube. Pipe the buttercream onto each cookie and then top with another gingerbread cookie. If you don't have a pastry bag, just spoon a dollop of buttercream onto each cookie.