How to Make It
Prepare the cake: Preheat oven to 325°F. Spray a 13- x 9-inch baking pan with baking spray. Beat sugar, butter, oil, and vanilla with an electric mixer on high speed until fluffy, about 2 minutes. Add eggs; beat until incorporated, about 15 seconds.
Stir together flour, cocoa powder, baking powder, baking soda, and salt in a bowl. Add flour mixture to butter mixture alternately with sour cream, beginning and ending with flour mixture, beating well on low speed after each addition. Stir in zucchini and chocolate chips. Pour into prepared pan; smooth top with a spatula.
Bake in preheated oven until a wooden pick inserted in center comes out clean, 45 to 50 minutes. Remove from oven; cool completely in pan, about 1 hour.
Prepare the frosting: Melt butter in a small saucepan over medium-high. Cook, stirring occasionally, until butter begins to brown, about 10 minutes. Remove from heat; pour into a heatproof glass bowl. Chill 1 hour.
Transfer solidified brown butter to bowl of a heavy-duty stand mixer fitted with a paddle attachment; beat on medium-high speed until creamy, about 30 seconds. Add cream cheese; beat on high speed until smooth, about 20 seconds. Reduce speed to low; gradually beat in powdered sugar, vanilla, and salt. Increase speed to high, and beat until fluffy, 2 minutes. Stir in 5 tablespoons of the pistachios.
Spread frosting over cake. Sprinkle with remaining 3 tablespoons pistachios.