In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma. Once browned and nutty, remove from the heat and pour into a shallow dish. Chill in the freezer for about 45 minutes, or until solid.

In a large mixing bowl, whisk together the flour, cream of tartar, baking soda, and salt. Set aside.

In the bowl of a stand mixer fitted with a paddle attachment, beat the solidified browned butter until smooth, about 1 to 2 minutes. Add both sugars and continue beating on medium speed until smooth and creamy. Scrape down the sides of the bowl, add the egg, and beat just until combined. Stir in the milk and vanilla. Add the flour mixture and beat on low just until combined. Dough will seem rather sticky, but that's normal. Cover dough with plastic wrap and refrigerate for at least 2 hours, or up to 5 days. Do not skip this step.

Preheat oven to 375ºF degrees and line a baking sheet with parchment paper or spray with nonstick cooking spray. In a small bowl, combine the sugar and cinnamon.

Using your hands or a small cookie scoop, form dough into 1-inch balls, roll in the cinnamon sugar, then place on the lined baking sheet, at least 2 inches apart. Bake for 8-9 minutes or until edges have set and the tops appear slightly undercooked. Cookies will firm up as they cool. Allow cookies to cool on the baking sheet for at least 5 minutes before transferring to a cooling rack to cool completely.

Brown the butter the same as you did with the cookie dough. Pour butter into a shallow dish and freeze until solid, about 45 minutes. Once solid, cream butter in a stand mixer fitted with the paddle attachment for about 2 minutes. Add the powdered sugar, a cup at a time, beating until combined. Add the milk and vanilla and mix on high until light and fluffy.

Spoon the buttercream into a piping bag fitted with a large round tip (or a large zip-loc bag with the tip cut off), and pipe the buttercream onto the bottom side of a cookie. Gently press another cookie on top, forming a sandwich. Roll the sides of the cookie sandwich in the leftover cinnamon-sugar coating. Repeat with the remaining cookies.