First, brown the butter for the dough. Melt ½ cup (113g) of butter in a light-colored saucepan over medium heat. Continue cooking as it foams, swirling the pan, until brown specks form and it smells nutty. Immediately pour into a heatproof bowl to cool until solidified, about 20-30 minutes in the refrigerator.
While the butter cools, make the cinnamon filling. In a small bowl, combine 2 ½ tablespoons (35g) softened butter, ¼ cup (55g) brown sugar, 1 ½ teaspoons (4g) cinnamon, 1 ½ tablespoons (12g) flour, and ⅛ teaspoon salt. Mix to form a paste. Roll into tiny ½-teaspoon balls, place on a parchment-lined plate, and freeze for at least 15 minutes.
Preheat your oven to 350°F (175°C). Line a large baking sheet with parchment paper.
In a large bowl, use an electric mixer to beat the cooled browned butter, ½ cup (100g) granulated sugar, and ¼ cup (55g) brown sugar until fluffy, about 3 minutes. Beat in the egg and 1 ½ teaspoons (7ml) vanilla extract until just combined.
In a separate bowl, whisk together 1 ½ cups (180g) flour, ½ teaspoon baking soda, ¼ teaspoon baking powder, ½ teaspoon salt, and ⅛ teaspoon nutmeg. Add the dry ingredients to the wet and mix on low speed until just incorporated. Do not overmix.
Gently fold the frozen cinnamon bits into the dough. Use a 3-tablespoon scoop to form dough balls and place them 2-3 inches apart on the prepared baking sheet. You can press a reserved cinnamon bit onto the top of each cookie if you like.
Bake for 10-12 minutes, until the edges are light golden brown. Let the cookies cool on the baking sheet for 5 minutes, then move them to a wire rack to cool completely.
Make the glaze. In a small bowl, whisk together ½ cup (60g) powdered sugar, 1 tablespoon (15ml) milk, and ¼ teaspoon vanilla extract until smooth. Drizzle the glaze over the cooled cookies and allow it to set before serving.