Preheat the oven to 350°F. Line sheet trays with parchment paper, set aside.
In a medium-sized bowl, whisk together the flour, cinnamon, baking soda, and salt, set aside.
In a large bowl, using a hand mixer cream the butter with the sugars until fully combined, 2 minutes.
Add the egg and vanilla, mix in well. Add half of the flour mixture, mix until combined.
Add the remaining flour and mix until combined. Scrape down the sides as needed.
Add the oats and mix in fully.
Dollop rounded tablespoon-sized cookie dough balls onto the sheet trays, 2 inches apart. I use a 1 tablespoon-sized cookie scoop for this. Try to get an even number of cookies so you can make sandwiches.
Bake for 9-10 minutes until lightly golden brown along the edges. Let the cookies sit on the tray for 2 minutes, transfer cookies to a wire rack to cool completely.
While the cookies cool, make the filling by adding the butter to a large bowl, with a hand mixer whip until smooth. Add the powdered sugar and slowly mix in until combined.
Add the milk and vanilla, mix until combined then turn the speed on medium-high and whip the filling until fluffy, 2 minutes.
Take half of the cookies and turn them upside down. Pipe the filling onto the cookie, leaving a quarter of an inch border along the edge. Place another cookie, top side up on top of the filling.
Gently press down so the filling disperses to the edge. Serve immediately.