Combine the cake mix, eggs, vegetable oil and 1 cup of water in a large bowl. Stir in half of the clementine zest and juice. Pour into the prepared pans. Bake until a cake tester comes out clean, 30 to 35 minutes. Let the cakes cool in the pans for 5 minutes, then invert onto a rack and cool completely. While the cakes cool, make the frosting.
In a stand mixer fitted with a whisk attachment, beat the butter until light and fluffy, about 2 minutes. Add half of the powdered sugar and carefully mix until thoroughly incorporated. Add the remaining powdered sugar and mix again until fully combined. Add the remaining clementine juice and zest, then beat until light and fluffy, about 2 more minutes. Add the food coloring to achieve your desired color, starting with a few drops and adding until you get the color you like.
Place 1 cooled cake layer on a cake stand or plate. Top with about 1/2 cup frosting and spread evenly. Top with the remaining cake layer. Spread the remaining frosting all over the top and sides of the cake. Sprinkle the top and sides of the cake with sprinkles.
Take the thin slices of clementine and make a slit halfway through. Twist the slices and place on the top of the cake.