Preheat oven to 350 F. Line baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, add butter, brown sugar, and sugar. Beat until light and fluffy.
½ cup salted butter, 1 cup light brown sugar, ¼ cup granulated sugar
Add eggs and vanilla. Beat again until combined, scraping down the sides of the bowl as needed.
2 large eggs, 1 teaspoon vanilla extract
Add oats, flour, cinnamon, nutmeg, baking powder, baking soda, and salt. Mix in until just combined.
2 cups old fashioned rolled oats, 1 ¾ cups all purpose flour, 1 ½ teaspoons ground cinnamon, ¼ teaspoon ground nutmeg, ¾ teaspoon baking powder, ½ teaspoon baking soda, ½ teaspoon kosher salt
Drop 1 ½ tablespoons cookie dough per cookie, about 2 inches apart on the prepared baking sheets.
Bake for about 10-15 minutes, or until light golden brown. Allow to cool on the baking sheet.
Once cooled, prepare the filling. In the bowl of your stand mixer, fitted with the paddle attachment, beat butter for about 5 minutes until fluffy.
¾ cup salted butter
Add in the powdered sugar and beat until smooth and combined.
3 cups powdered sugar
Add milk and vanilla. Beat again until combined.
3 tablespoons milk, 1 ½ teaspoons vanilla extract
To assemble the cookies, pipe frosting on the back of one cookie, then top with another. Repeat until all cookies are sandwiched with filling in the center.
Notes
Butter: If using unsalted butter, increase the salt in the cookies to 3/4 teaspoon and add 1/4 teaspoon salt to the filling.
Brown sugar: I use light brown sugar, but dark brown sugar will also work.
Flour: To properly measure the flour, either weigh it (click "metric" just above the ingredients for the weights) or sift/stir to break up the flour, lightly spoon into the measuring cup, and level.
Salt: If you don't have kosher salt or sea salt, use half the amount of table salt.