First, make the cookies: Heat oven to 325°F. Line 2 baking sheets with parchment paper.
Add flour, 1 cup powdered sugar and salt to a food processor. Pulse a few times to combine, then add maple extract and 1 cup cubed butter. Pulse mixture until butter is incorporated and mixture looks like wet sand and begins to climb up sides of bowl.
Transfer dough to a counter and gently press into a mound. The mixture will be loose and crumbly, but knead it a few times until it comes together into a cohesive shape. Gently form dough into a square. Sprinkle flour onto counter next to the dough; place dough on top of floured surface.
Using a well-floured rolling pin, roll dough out to 1/4-inch thickness. Using a floured 2-inch round cookie cutter, cut dough into shapes (you can also use a different cookie cutter shape or cut the dough into squares, if desired). Re-roll dough out 1-2 times to cut out more shapes, if needed. You should end up with approximately 36-40 cookies.
Transfer cookies to one lined baking sheet. Place baking sheet in freezer and freeze cookies for about 15 minutes or until cookies are firm.
Transfer some of the cookies to second lined baking sheet -- as many as can fit spaced 1/2-inch apart (keep remaining cookies in freezer). Bake cookies 15 to 18 minutes until baked through and edges are lightly golden.
Cool cookies 1-2 minutes on cookie sheet, then transfer to a cooling rack to cool completely. Repeat step 6 as needed until all cookies are baked.
While cookies cool, make the maple buttercream frosting: In a large bowl using an electric hand mixer or in the bowl of a stand mixer fitted with the paddle attachment, beat remaining 1/2 cup butter on medium speed until smooth. Gradually beat in remaining 2 cups powdered sugar until smooth. Beat in maple syrup. Increase speed to medium-high and beat frosting 1-2 minutes or until fluffy and light.
On a work surface, lay half of cooled cookies bottom side-up. Use a piping bag, storage bag with one tip snipped off or a spatula to pipe or spread approximately 1 generous tablespoon frosting on cookies. Top with remaining cookies to form a sandwich.
Store cookies in an airtight container at room temperature or refrigerated up to 4 days.