Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, cream together butter, brown sugar, and granulated sugar until light and fluffy.
Beat in eggs one at a time, then stir in vanilla extract.
In another bowl, whisk together flour, baking soda, cinnamon, and salt. Gradually add to the creamed mixture. Stir in oats.
Drop dough by rounded tablespoonfuls onto prepared baking sheets. Flatten slightly with the back of a spoon.
Bake for 10-12 minutes, or until edges are golden. Cool on baking sheets for 5 minutes, then transfer to wire racks to cool completely.
For the filling, beat together butter and shortening until smooth. Gradually add powdered sugar, beating until light and fluffy. Add eggnog and nutmeg, and beat until well combined.
Spread filling on the flat side of half the cookies. Top with remaining cookies to form sandwiches.