Combine the flour, baking soda, and salt in a medium bowl. In the bowl of an electric mixer, beat butter, maple syrup, and brown sugar on high until light and fluffy, about 3 minutes. Blend in the egg. Gradually add the flour mixture, beating well between each addition.
Divide the dough in half and flatten into two disks. Wrap in wax paper and refrigerate until firm, about 1 hour.
Preheat oven to 350 degree F. Line 3 cookie sheets with parchment paper. On a floured work surface, roll out one disk until it's about 1/4 inch thick. Use a 1-1/2-inch-diameter glass or round cookie cutter to cut out as many circles from the dough as possible. Transfer the circles to cookie sheets, about 1 inch apart. Press a smaller glass or cookie cutter partway into the dough to create the button's rim. Use a straw or a toothpick to make the four holes in the center of each cookie.
Bake until golden, 10 to 12 minutes. Transfer to a wire rack and let cool completely. When ready to serve, thread licorice through buttonholes.