In a small saucepan, melt butter over medium-high heat. Cook, stirring frequently, until butter turns a medium-brown color and has a nutty aroma, about 10 minutes. Remove from heat, and pour into a heatproof bowl; let cool to room temperature and solidified. In a small microwave-safe bowl, reserve 2 tablespoons (28 grams) browned butter.
Preheat oven to 350°F (180°C). Line baking sheets with parchment paper.
In the bowl of a stand mixer fitted with the paddle attachment, beat brown sugar and remaining browned butter at medium speed until light and fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Beat in egg until well combined. Beat in vanilla.
In a medium bowl, whisk together flour, cornstarch, salt, baking powder, and baking soda. With mixer on low speed, gradually add flour mixture to sugar mixture, beating until just combined, stopping to scrape sides of bowl. Using a 1½-tablespoon spring-loaded scoop, scoop dough (about 30 grams each), and place 1½ inches apart on prepared pans.
Bake until edges are golden brown, 12 to 15 minutes. Let cool on pans for 5 minutes. Remove from pans, and let cool completely on wire racks.
Microwave reserved browned butter on high until melted; whisk in powdered sugar and milk until smooth and well combined. Drizzle onto cooled cookies. Let stand until glaze is set, about 1 hour. Store in an airtight container for up to 5 days.