Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper and set aside.

In a large mixing bowl, beat the softened butter, brown sugar, and granulated sugar with an electric mixer until the mixture is light and fluffy. Add the eggs and egg yolks, one at a time, beating well after each addition. Stir in the vanilla extract and buttermilk.

In a separate bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Gradually add the dry ingredients to the butter mixture, stirring gently until just combined and no visible flour remains. Do not overmix.

Using a spoon or cookie scoop, drop 2-tablespoon portions of dough onto the prepared baking sheets, spacing them approximately 2 inches apart. Bake for 10 to 12 minutes, or until the edges are lightly browned. Allow the cookies to cool on the baking sheets for 10 minutes before transferring them to wire racks to cool completely.

In a medium saucepan over low heat, melt the butter and brown sugar together, stirring constantly until smooth. Stir in the salt and buttermilk, and remove the pan from heat. Gradually whisk in the sifted powdered sugar, 1/4 cup at a time, until the glaze reaches the desired consistency. More powdered sugar may be added if a thicker glaze is preferred.

Once the cookies are completely cooled, spread or drizzle the buttermilk glaze evenly over the tops. Allow the glaze to set before serving.