How to Make It
Preheat the oven to 350°. Oil and flour a 10-inch springform pan.
Arrange the walnuts on a baking sheet and toast in the oven, shaking once or twice, until golden-brown, about 10 minutes. Remove from the oven and cool completely. Chop finely or grind in a nut grinder.
In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon, cloves and salt.
Beat together 1/2 cup olive oil and 1/2 cup sugar until creamy. Add the egg yolks and yogurt and beat until well combined. Beat in the flour mixture and the ground walnuts.
In a separate bowl, combine the egg whites with the cream of tartar and beat with a whisk attachment until soft peaks form. Gradually add the remaining 1/2 cup sugar and beat until the egg whites are stiff and hold glossy peaks. Gently but thoroughly fold the egg whites into the batter. Spoon the batter in the prepared cake pan and smooth the top with a spatula.
Bake until the top of the cake is golden and a wooden skewer inserted in the center of the cake comes out clean, 45 to 50 minutes.
Cool the cake in the pan on a rack for 10 minutes, then remove the side of the pan and cool completely.
Bring the sugar and juices to a simmer in a small saucepan and simmer, stirring, until the sugar is dissolved. Continue to simmer until syrup is slightly thickened and reduced to about 1 cup. Remove from the heat and let cool.
Cut the peel and pith from the oranges with a sharp paring knife. Cut segments free from the membranes, letting them drop into a bowl. Pour the cooled syrup over the oranges.
Serve the cake topped with some of the oranges and a generous drizzle of syrup.