Spiced Carrot Cake with Brown Butter Frosting

Melt butter in a large, heavy-bottomed saucepan over medium-high, stirring often; gently simmer butter until white milk solids separate from butter-fat, about 10 minutes. Reduce heat to low, and cook, stirring constantly, until butter is golden brown and spoon is covered in flecks of caramelized milk solids, about 2 minutes. Remove from heat. Line a fine wire-mesh strainer with 3 layers of cheesecloth, and place strainer over a heatproof container. Pour melted butter through strainer and into container. Reserve caramelized milk fat (about 1 teaspoon), and set aside. Let strained ghee cool completely at room temperature, about 1 hour.

Preheat oven to 350°F. Coat a 13- x 9-inch baking pan with cooking spray; set aside. Combine brown sugar and orange zest in a large bowl. Rub orange zest into sugar using your fingers. Add eggs, egg yolk, vanilla, cinnamon, cardamom, ginger, nutmeg, and cooled brown butter ghee. Beat with an electric mixer on high speed until pale and fluffy, 2 to 3 minutes. Fold in shredded carrots, pistachios, and raisins until well combined. Whisk together flour, baking powder, baking soda, and salt in a medium bowl until combined. Fold flour mixture into carrot mixture, stirring until just combined. Pour batter into prepared pan; spread in an even layer.

Bake carrot cake in preheated oven until a wooden pick inserted in center comes out clean, about 35 minutes. Remove from oven. Let cool completely in pan, about 1 hour.

Beat butter, cream cheese, vanilla, and reserved caramelized milk fat with a stand mixer fitted with the paddle attachment on medium-high speed until smooth, about 1 minute. Reduce mixer speed to low; gradually beat in powdered sugar until combined. Increase mixer speed to medium-high, and beat until pale and fluffy, 2 to 3 minutes.

Spread frosting onto cooled cake in an even layer. Garnish with finely chopped pistachios and shredded carrots, if desired.