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Cook butter in a heavy saucepan over medium heat, stirring constantly until it foams, turns clear, and then turns a deep brown, about 6 minutes. Pour browned butter into a glass measuring cup and chill in the freezer until solidified, about 45 minutes to an hour.
Whisk together flour, cornstarch, baking soda, cream of tartar, and salt in a medium bowl. Beat solidified brown butter and sugars in a large mixer bowl at medium speed until creamy; beat in egg and egg yolk. Add maple extract and maple syrup, beating until well combined. Reduce speed to low and gradually add flour mixture; beat just until incorporated. Cover dough and chill in the refrigerator for at least 4 hours, or overnight.
Preheat oven to 350°F (If cookie dough has chilled overnight, let stand at room temperature for 20 minutes before baking). Line two baking sheets with silicone mats or parchment paper. Form 1 1/2-inch balls of dough and place two inches apart on prepared baking sheets; flatten mounds slightly. Bake, rotating pans halfway through, for 11 to 12 minutes, until lightly golden and edges are set. Cool for 3 minutes on baking sheets, then transfer to a wire rack to cool completely.
For the glaze, whisk together melted butter, maple syrup, and salt in a small bowl. Whisk in powdered sugar, milk, and maple extract until smooth. Drizzle glaze over cookies; let set before serving.