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

In a large bowl, beat together softened butter, brown sugar, and granulated sugar for 2 to 3 minutes until light and fluffy.

Add the egg and vanilla extract to the butter mixture and beat until smooth, scraping down the sides of the bowl.

In a separate bowl, whisk together flour, rolled oats, baking soda, salt, and cinnamon until evenly combined.

Add the dry ingredients to the wet ingredients and stir until just combined, being careful not to overmix.

Using a cookie scoop or spoon, drop dough balls about 2 inches apart onto the prepared baking sheets.

Bake cookies at 350°F (175°C) for 11 to 13 minutes, until the edges are golden brown and the centers still look slightly underdone.

Let cookies cool on the baking sheets for 5 minutes, then transfer to wire racks and cool completely, about 30 minutes.

In a clean bowl, beat softened butter and vegetable shortening until smooth, then add powdered sugar, vanilla extract, and salt and beat on medium speed for 3 to 4 minutes until light and fluffy.

Add milk one tablespoon at a time and beat gently until the filling is thick but spreadable.

Pair cookies of similar size and spread or pipe a generous amount of filling onto the flat side of one cookie, then top with the second cookie to form a sandwich.