Olive Oil Cake with Buttercream Frosting and Sour Candied Oranges

Preheat oven to 350°F. Spray a 9x13-inch baking dish with cooking spray; line bottom with parchment paper and spray paper. Using an electric mixer, beat sugar and eggs until pale yellow, 2 to 3 minutes. While eggs and sugar are mixing, whisk both flours, ginger, salt, baking powder, baking soda, and cloves in a medium bowl. With machine running, gradually add oil to egg mixture and beat until smooth, then gradually beat in milk. Turn off machine and add dry ingredients. Mix on low just to blend. Increase speed to medium and mix until smooth, stopping once or twice to scrape down sides and bottom of bowl, about 2 minutes. Pour batter into prepared pan.

Bake cake until a tester comes out clean when inserted into center of cake, about 50 minutes. Let cool in pan 10 minutes. Run a knife around edges of pan to loosen, then invert onto a wire rack. Remove parchment paper and let cool completely.

Fill a medium saucepan halfway with water and bring to a boil. Add citrus slices and cook for 2 minutes. Using a slotted spoon, transfer slices to a wire rack set inside a rimmed baking sheet. Repeat cooking and draining two more times, using a fresh pot of boiling water each time. Remove any seeds from slices.

Add 1/2 cup water, 1/2 cup sugar, honey, and corn syrup to same medium saucepan and stir to blend. Add citrus slices and bring to a simmer over medium, gently stirring occasionally, until sugar melts. Reduce heat to low and cook, occasionally stirring gently and keeping citrus slices intact, until citrus rinds begin to turn translucent, about 30 minutes. Transfer to a wire rack and let cool for 15 minutes. Reserve 2 tablespoons of the syrup for cake, and refrigerate remainder of syrup to flavor soda, coffee, tea, or cocktails.

In a wide shallow bowl, whisk remaining 1/2 cup granulated sugar, citric acid, and sea salt to evenly distribute. Coat citrus slices on both sides with sugar mixture, pressing to adhere. Return to rack and let dry slightly while cake cools.

Add butter and vegetable shortening to bowl of a stand mixer fitted with a paddle attachment. Beat on low speed until smooth and glossy, stopping once to scrape down sides and bottom of bowl, about 1 minute. Add powdered sugar and mix on low to blend. (Cover bowl with a towel to avoid a big mess.) The mixture will look crumbly at first, but will come together with more mixing.

Increase speed on mixer to high and mix until buttercream is fluffy and pale, stopping once to scrape down sides and bottom of bowl, about 2 minutes. The longer you mix the buttercream, the whiter it will become. If you'd like, add milk to buttercream, 1/2 tsp at a time, until buttercream is smooth and spreadable. Beat in salt and vanilla or lemon zest, if using. Buttercream can be stored airtight at room temperature for up to a week. Mix again in stand mixer using a paddle attachment before using.

Brush top of cake with reserved 2 tablespoons syrup. Spread buttercream over top and sides of cake. Garnish top with citrus slices and edible flowers and leaves.