Brown the butter in a saucepan over medium heat until nutty and amber, 5–7 minutes. Cool 10 minutes.
Whisk flour, baking powder, baking soda, salt, and 2 tsp cinnamon in a bowl.
Beat cooled brown butter with granulated and brown sugars until sandy and combined.
Add beaten eggs and vanilla. Mix until glossy and slightly thick.
Stir in dry ingredients just until no dry streaks remain. Chill dough 30 minutes.
Heat oven to 350°F. Line two baking sheets with parchment.
Combine 1/4 cup sugar with 1 1/2 tsp cinnamon in a shallow bowl. Place powdered sugar in another bowl.
Scoop dough into 1 1/2-tbsp portions. Roll into balls, then coat in cinnamon sugar and generously in powdered sugar.
Arrange 2 inches apart. Bake 10–12 minutes until crackled with set edges and soft centers.
Cool on the sheet 5 minutes, then move to a rack to finish cooling.