Zest all 5 blood oranges and divide between 3 separate small bowls: 2 bowls should contain the zest of 1 orange each, and 1 bowl should contain the zest of 3 oranges. Set aside the 2 bowls with less zest for later use.
Add the olive oil to the bowl with the zest of 3 oranges. Set aside for 30-60 minutes to infuse.
Juice 3 of the oranges and set aside 3 tablespoons of juice for the glaze. Reserve the remaining orange juice in a liquid measuring cup. You should have about ½ cup. (120 ML)
Preheat the oven to 350˚F (180˚F).
In a large bowl, sift together the flour, cocoa powder, baking soda, and salt.
Add the sugar, blood orange olive oil (with the zest),½ cup (120 ml) of the blood orange juice, water, and white vinegar. Whisk until the batter is smooth with no lumps.
Lightly grease the bottom and sides of an 8-9-inch (20-22 cm) round cake pan with olive oil, and place a parchment round in the bottom of the pan. (To make the parchment round, tear off an 18 x 13-inch (45x33 cm) piece of parchment paper. With a long edge closest to you, fold down the top right corner to meet the bottom edge. Press down on the long side of the triangle to flatten, then turn so that side is facing you and fold in half, bringing the points together. Turn again so the long side of the new triangle is on top and fold in half twice more. Flip the cake pan upside down and hold the pointy end of the triangle at the center of the pan. Trim the excess parchment that extends past the edge of the pan. Flip the pan, unfold the triangle, and place the parchment round in the bottom).
Pour the batter into the prepared pan and bake for 35-40 minutes, until a toothpick inserted in the center of the cake comes out clean.
While the cake bakes, prepare the toppings: Cut the top and the bottoms off the 2 remaining zested oranges. Using a paring knife, carefully remove the skin and pith from the oranges until you are left with just the flesh. Turn each orange on its side and cut into ¼-inch (6 mm) wide rounds. Place the rounds between 2 paper towels to blot some of the excess juice. This will allow them to stick to the glaze more effectively.
Make the glaze: Sift the powdered sugar into a medium bowl. Add the reserved zest of 1 orange, and the remaining 3 tablespoons of blood orange juice. Whisk into a glaze. It should be thick, but spreadable.
Remove the cake from the oven. Let cool in the pan for 10 minutes, then flip onto a wire rack set over a baking sheet to cool completely.
Once the cake has cooled, pour the glaze over the cake, letting it drip down the sides. Decorate the top with the orange slices, pistachios, and remaining orange zest.