Preheat oven to 350°F. Butter three 8-in. round cake pans; line bottoms with parchment. Set aside. Sift flour and baking powder in a bowl; whisk in salt. Set aside.
Beat sugar, butter, and coconut oil in a stand mixer fitted with a paddle attachment on medium-high speed until fluffy, 3 to 4 minutes. Add eggs 1 at a time, beating well after each addition. Reduce mixer speed to low; beat in sour cream and vanilla to combine. Gradually add flour mixture, and beat until almost combined, about 1 minute. Divide batter evenly among three bowls.
Add carrot juice and 1/4 cup of the milk to one bowl of batter; fold together until smooth and just combined. Add beet juice and 1/4 cup of the milk to second bowl of batter; fold together until smooth and just combined.
Process mint and remaining 1/2 cup milk in a food processor until smooth, 2 to 3 minutes; add mixture to third bowl of batter, and fold together until smooth and just combined. Transfer each batter to a prepared cake pan, spreading evenly in pans.
Bake until edges of the layers are lightly browned and a wooden pick inserted into centers comes out with a few crumbs, about 30 minutes. Let cool 10 minutes. Remove layers from pans, and transfer to a wire rack. Let cool completely, about 1 hour.
Beat butter and cream cheese in a stand mixer fitted with a paddle attachment on medium speed until creamy, about 1 minute. Reduce mixer speed to low, and gradually beat in confectioners' sugar, whipping cream, vanilla, and salt. Increase mixer speed to medium, and beat mixture until creamy, 1 to 2 minutes.
Using a serrated knife, trim domed tops of the cake layers to make flat. Place 1 layer on a platter; spread top evenly with some of the frosting. Top with second layer; spread top evenly with some of the frosting. Top with remaining layer; spread top and sides of cake with remaining frosting. Decorate with sprinkles.