Add the unsalted butter to a saucepan and melt over medium heat. Continue cooking, stirring occasionally, until the butter becomes golden, smells nutty, and brown specks appear on the bottom (about 5–7 minutes). Remove from heat and let the browned butter cool for 10 minutes.
In a large mixing bowl, whisk together the cooled browned butter, dark brown sugar, and granulated sugar until smooth. Add the egg, vanilla bean paste, and milk. Whisk again until fully combined.
Add the flour, baking powder, baking soda, and salt. Stir just until the dough forms—do not over-mix. Fold in the chopped milk chocolate and ⅓ cup mini holiday M&Ms.
Scoop the dough into balls 12 equal cookie dough balls and roll them until smooth. Dip the tops of each dough ball into the extra mini holiday M&Ms so they stick to the surface.
Place the dough balls on a tray lined with tin foil or parchment paper and freeze for at least 30 minutes. This prevents spreading and helps the cookies bake thick and soft.
Preheat the oven to 350°F once the cookies have been in the freezer for about 20 minutes.
Arrange the chilled cookie dough balls on a baking sheet lined with parchment. Bake for 12–16 minutes, or until the edges are lightly golden and the centers still look slightly soft.
Allow the cookies to cool on the baking sheet for 10 minutes, then transfer them to a cooling rack to cool completely.