1 1/2 sticks (3/4 cup) unsalted butter, cubed, at room temperature
2 teaspoons pure vanilla extract
2 large eggs, lightly beaten
2 1/2 cups all-purpose flour, plus more for dusting if needed (see Cook’s Note)
1/4 teaspoon baking powder
1/4 teaspoon kosher salt
3/4 cup red, green and white sprinkles
3/4 cup confectioners’ sugar
1 tablespoon whole milk
Beat the granulated sugar and butter in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 5 minutes. Add the vanilla and eggs, then continue to mix until incorporated (it’s ok if the mixture looks slightly curdled). Scrape down the sides of the bowl with a rubber spatula as needed.
Meanwhile, whisk together the flour, baking powder and salt in a medium bowl until evenly combined. Reduce the mixer to medium low, then slowly add the flour mixture and mix until just combined.
Divide the dough into 18 portions (about 2 1/2 tablespoons each), then roll between the palms of your hands into uniform balls; place on a parchment-lined baking sheet. Pour the sprinkles into a shallow bowl or baking dish. Roll the dough balls in the sprinkles until completely coated, then return to the baking sheet. Refrigerate until firm and chilled through, about 1 hour.
Position oven racks in the top and bottom thirds of the oven and preheat to 350 degrees F. Line one additional baking sheet with parchment.
Place 9 chilled dough balls on each prepared baking sheet. Using a 1/3-cup measuring cup, smash the dough balls down until the edges are starting to crack and they are 3 to 3 1/2 inches wide (leave about 1/2 inch of space between each smashed cookie). If the dough starts to stick to the bottom of the measuring cup, dust with a little flour and continue smashing.
Bake, rotating the baking sheets halfway through, until the cookies puff slightly, the centers are still soft and the bottoms are just golden brown, 14 to 17 minutes. Let cool on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely.