Preheat the oven to 350°F (175°C). Line two large baking sheets with parchment paper and set them aside for the shaped cookie dough.
Place the 2 1/3 cups all-purpose flour, 1 teaspoon baking soda, and 1/2 teaspoon kosher salt in a medium mixing bowl. Whisk thoroughly until the leavening and salt are evenly distributed throughout the flour.
Add the softened 1/2 cup unsalted butter and 1 cup packed light brown sugar to the bowl of a stand mixer fitted with the paddle attachment. Beat at medium speed for approximately 1–2 minutes, or until the mixture becomes smooth, pale, and creamy.
Add the large room-temperature egg and continue mixing until it is fully incorporated into the butter and sugar mixture.
Pour in the 1/3 cup pure maple syrup, 1 teaspoon vanilla extract, and 1 teaspoon maple extract. Beat on high speed until the ingredients are evenly blended, stopping as necessary to scrape down the sides and bottom of the mixing bowl.
Add the previously prepared dry ingredients. Mix on low speed only until the flour has disappeared into the dough and no dry streaks remain. Avoid excessive mixing so the finished cookies retain a tender texture.
Using a 2-tablespoon cookie scoop, portion the dough and gently roll each piece into a ball. Arrange the dough balls on the parchment-lined baking sheets, leaving approximately 2 inches of space between each cookie.
Bake for 8–9 minutes, until the edges have developed a light golden-brown color while the centers still appear slightly underbaked.
Allow the cookies to remain on the hot baking sheets for 5 minutes. Transfer them carefully to a wire cooling rack and leave them until completely cooled.
While the cookies cool, place the 1/2 cup sifted confectioners’ sugar, 3 tablespoons pure maple syrup, 1 tablespoon melted and cooled unsalted butter, 1/4 teaspoon maple extract, and a pinch of kosher salt in a medium bowl.
Stir until the icing is completely smooth and has a thick yet pourable consistency. The maple syrup may be incorporated gradually to help control the final thickness of the icing.
Once the cookies have cooled fully, drizzle the maple icing evenly across their tops.
Allow the icing to set before serving or storing the cookies.