Make the blood orange curd. Whisk together egg with egg yolks, sugar, blood orange juice, and zest until well combined. Place bowl over double boiler and cook until thickened, about 10 minutes, whisking frequently. Remove from heat and whisk in butter, mixing until fully incorporated. Place plastic wrap directly on surface of curd and let cool. Chill in the fridge until ready to use.
Preheat oven to 350 degrees. Line and grease two 9-inch cake pans with cooking spray.
Make carrot cake. Whisk together eggs with both sugars, olive oil, applesauce, vanilla, and orange zest until well combined.
In a separate bowl, combine flour with baking powder, baking soda, cinnamon, cardamom, ginger, nutmeg, and salt until well combined. Add dry ingredients to wet ingredients, stirring just until combined. Fold in grated carrots and chopped pistachios. Divide cake batter between cake pans.
Bake for 30-35 minutes or until a toothpick inserted in the center comes out clean. Let cakes cool in pan for 25 minutes. Invert onto wire racks and let cool completely.
Prepare frosting. Beat together butter with cream cheese in the bowl of a stand mixer until well combined. Add powdered sugar and vanilla and beat until well combined.
Place bottom cake on cake stand. Pipe edges of cake with frosting to create a border. Fill the center with blood orange curd, about ¾ cup. Stack remaining cake on top and spread frosting on top. Serve at room temperature.