Preheat oven to 350°F (175°C).
In a large bowl, cream together the butter, brown sugar, and granulated sugar until light and fluffy.
Beat in the eggs one at a time, then stir in the vanilla.
Combine the flour, baking soda, and salt; stir into the creamed mixture until just blended. Mix in the quick oats.
Drop by rounded teaspoonfuls onto ungreased baking sheets.
Bake for 8 to 10 minutes in the preheated oven, until the edges are golden. Allow cookies to cool on baking sheet for 5 minutes before transferring to a wire rack to cool completely.
To make the filling: In a medium bowl, cream together the butter, shortening, and vanilla. Gradually beat in the powdered sugar. Add milk one tablespoon at a time until the filling is spreadable.
Spread filling between two cookies to make sandwiches.