Cardamom-Pistachio Carrot Cake

Remove eggs from water and crack into a large bowl; add 1 cup (packed; 200 g) dark brown sugar, ¼ cup (50 g) granulated sugar, 1¾ tsp. baking powder, ¼ tsp. baking soda, 1¼ tsp. Diamond Crystal or ½ tsp. Morton kosher salt, 1½ tsp. ground cardamom, and ½ tsp. ground cinnamon. Using an electric mixer on high speed, beat 3 minutes (mixture should be thick, pale, and slightly increased in volume). Decrease speed to medium-high and gradually stream in reserved brown butter (including browned bits); beat until combined. Go slow during this part—if you add the butter too quickly, it'll be hard to achieve the emulsion you're aiming for.