Place 14 tablespoons (1 ¾ sticks) unsalted butter in the bowl of a stand mixer. (Alternatively, place in a large bowl if using an electric hand mixer.) Let sit at room temperature until softened.

Arrange a rack in the middle of the oven and heat the oven to 350°F. Line a baking sheet with parchment paper. Meanwhile, make the cookie dough.

Place ½ cup sliced almonds in a medium skillet over medium heat. Toast, stirring frequently, until the almonds darken in color slightly and are very fragrant, 6 to 8 minutes. Immediately transfer to the bowl of a food processor fitted with the blade attachment and let cool for 10 minutes. Process until the almonds are the texture of coarse flour and no large pieces remain, about 30 seconds.

Add ⅓ cup powdered sugar, ¼ cup granulated sugar, 2 teaspoons vanilla extract, and 1 large egg yolk to the butter and beat with the paddle attachment on medium-high speed until light and fluffy, 1 to 2 minutes. Turn the mixer off and scrape down the sides of the bowl. Add the ground almonds, 2 cups all-purpose flour, and ½ teaspoon kosher salt, and mix on low speed until a dough forms, 1 minute.

Transfer the dough onto a lightly floured surface and roll out ¾-inch thick. Using a 2-inch round cutter (or a standard shot glass), cut out rounds. Transfer the rounds to the prepared baking sheet, spacing them about 1-inch apart (these cookies won't spread during baking). Press the dough scraps together into a ball and repeat rolling and cutting out rounds. (If the dough becomes too soft to work with, form it into a ball, wrap tightly in plastic wrap, and refrigerate for 15 minutes before rerolling.) Freeze the rounds for 5 minutes.

Bake until the cookies are light brown on the tops and edges, 12 to 14 minutes. Let the cookies cool on the baking sheet for at least 30 minutes. Meanwhile, place 1 cup granulated sugar and 1 ¾ cups freeze-dried berries (1 ounce) in a food processor fitted with the blade attachment. Process until the berries are very finely ground and the sugar is tinted a light pastel color, about 30 seconds. Transfer to a medium bowl.

Toss the cookies one by one in the sugar to coat them on all sides.