Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.

In a large bowl, cream together the softened butter, brown sugar, and granulated sugar until light and fluffy.

Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.

In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add this mixture to the wet ingredients, mixing until just combined.

Fold in the old-fashioned oats until evenly distributed.

Drop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart.

Bake for 10-12 minutes, or until the edges are golden brown. Allow to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.

In a medium bowl, beat together the softened butter, powdered sugar, vanilla extract, and milk until smooth and creamy.

Once the cookies have cooled completely, spread a generous amount of filling on the flat side of half the cookies.

Top with the remaining cookies, pressing gently to adhere.

Store the cookies in an airtight container for up to a week.