Brown the butter for cookies: In a light-colored saucepan over medium heat, melt ½ cup butter, swirling occasionally. Cook until it turns amber, brown specks form, and it smells nutty. Immediately remove from heat and pour into a large bowl. Let cool for 10-15 minutes.
Mix wet ingredients: To the cooled brown butter, add brown sugar and granulated sugar. Beat for 2-3 minutes until light and fluffy. Add egg, ½ cup maple syrup, and vanilla extract. Beat until smooth and glossy.
Combine dry ingredients: In a separate bowl, whisk together flour, baking soda, salt, and cinnamon until evenly distributed.
Combine mixtures: With mixer on low, gradually add dry ingredients to wet ingredients. Mix just until combined and no flour streaks remain. Do not overmix.
Scoop and bake: Preheat oven to 350°F (175°C). Drop rounded tablespoons of dough onto parchment-lined baking sheets, 2 inches apart. Bake for 10-12 minutes, until edges are just set and beginning to brown. Centers will look soft.
Cool cookies: Let cookies cool on baking sheets for 5 minutes, then transfer to a wire rack to cool completely before icing.
Make brown butter icing: In the same saucepan, melt ½ cup butter over medium heat. Cook, stirring frequently, until golden brown and nutty. Remove from heat.
Finish icing: In a medium bowl, combine hot browned butter with powdered sugar, 2 tablespoons maple syrup, and ¼ teaspoon salt. Mix until smooth. Adjust consistency if needed.
Ice the cookies: Once cookies are completely cool, drizzle or spread icing over the top. Let icing set at room temperature for about 30 minutes before serving.
