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

In a large bowl, cream together the butter, brown sugar, and granulated sugar until light and fluffy. Beat in the egg and vanilla extract.

In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add to the wet ingredients, mixing until just combined. Stir in the rolled oats.

Drop by rounded tablespoons onto the prepared baking sheets. Bake for 10-12 minutes, or until the edges are golden brown. Let cool on the baking sheets for a few minutes before transferring to a wire rack to cool completely.

In a medium bowl, beat the butter until light and fluffy. Gradually add the powdered sugar, beating until smooth. Beat in the milk and vanilla extract until light and creamy.

Spread a generous amount of cream filling onto the flat side of one cookie. Top with another cookie, flat side down. Repeat with remaining cookies and filling.

Serve immediately or store in an airtight container in the refrigerator.