Beat the butter, 1/2 cup confectioners' sugar, 1/3 cup maple syrup, 1/4 teaspoon salt and the maple extract in a large bowl with a mixer on medium speed until smooth and creamy, about 2 minutes. Reduce the mixer speed to low; add the flour in 2 batches and beat until just combined.
Turn out the dough onto a lightly floured surface and dust with flour. Gently knead a few times until soft but not sticky, lightly dusting the surface with more flour as needed. Pat into an 8-inch square (about 1/2 inch thick). Cut into 1-by-2-inch rectangles, flouring the knife if needed.
Spread the walnuts on a plate. Gently press each rectangle in the walnuts to coat. Arrange about 1 1/2 inches apart on 2 baking sheets. Refrigerate until firm, about 30 minutes.
Position racks in the upper and lower thirds of the oven; preheat to 325 degrees F. Bake, switching the pans halfway through, until the cookies are firm and lightly browned on the bottom, about 25 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Make the glaze: Mix the remaining 1/2 cup confectioners' sugar, 3 tablespoons maple syrup and a pinch of salt in a small bowl until smooth. Drizzle on the cookies. Let set, about 15 minutes.