1. Preheat oven to 200 degrees C Remove beet greens and scrub beets. Arrange beets in a small baking dish, add ½-inch cold water and cover tightly with foil. Bake until beets are fork tender, 50 - 60 minutes, depending on size. Use caution when lifting foil as the steam plume will be very hot. Drain hot water and set beets aside to cool. Under cold running water, rub peel off with fingers or use a peeler. Trim beets. Beets can be prepped ahead to this point and refrigerated until needed.
2. When ready to make the cake, preheat oven to 180 degrees C. Grease a 9 x 5-inch loaf pan and dust with cocoa powder or use a baking spray.
3. Grate enough of the cooked beets to make 1 ¼ cups. Reserve ¼ cup grated beets to color frosting.
4. In a small bowl, combine flour, cocoa powder, baking powder and soda, and sea salt.
5. In a large bowl, or the bowl of a stand mixer fitted with the paddle attachment, beat eggs and sugar on low speed for 4 minutes. Add grated beets and vanilla and mix until incorporated. With mixer still on low, fold in half the flour mixture, all of the olive oil, and then the remaining flour. Mix just until everything is well combined.
6. Spoon into prepared pan and slide into oven. Bake 65 - 75 minutes or until a wooden skewer inserted in the center comes out with just a few crumbs attached.
7. Cool cake on a wire rack for 20 minutes, then carefully unmold and set on wire rack to cool completely.
8. To make frosting, beat frosting on low speed in the bowl of a stand mixer fitter with the paddle attachment or a large mixing bowl for 3 minutes. Add the butter and beat 3 minutes more, scraping down sides as needed. Sift in powdered sugar, add sea salt, and beat 4 minutes longer, scraping down sides as needed. Add blood orange juice and zest, and beat until well incorporated. Add beet juice a little at a time, until the desired color is achieved.
9. Frost cooled cake, garnish with blood orange segments, and pink sprinkles. Cake and frosting can both be made ahead (frosting should be refrigerated); set cold frosting on counter to warm slightly before frosting cake.