Preheat oven to 375F. (See Notes, below.) Line a baking sheet with parchment.
Combine rolled oats, flour, maple flakes, baking soda, baking powder, and salt in a medium bowl. Whisk to combine.
In a large bowl, cream butter with brown sugar. Add egg and maple extract and mix well.
Add dry ingredients to butter and sugar mixture. Stir to combine,
Drop by tablespoon onto the prepared baking sheet. (See Notes, below.)
Gently press a pinch of maple flakes onto the top of the cookies, then put the pan in the oven, Bake for a about 4-5 minutes, turn the pan, then bake for 3-5 more minutes or until cookies are golden around the edges.
Remove cookies from oven and let rest on the pan for about 5 minutes, then move them to a wire rack to cool completely.
When cookies have cooled, make the icing. Add all icing ingredients to a small bowl, then whisk until smooth. If the icing is too thin, add more icing sugar, a teaspoon at a time. If it is too thick, add more maple syrup or a bit of whipping cream, a couple of drops at a time. (The icing has to have some thickness to stay in the cookie.)
Apply a layer of icing to the bottom of one cookie, then top it with another cookie to make a cookie sandwich. Store in an airtight container for 2-3 days.