In a medium saucepan on high heat, combine the sugar and the water. Bring to a boil. Once boiling, reduce the heat to low and carefully add in the orange slices, making sure the syrup mostly covers them all. Let simmer for 15 minutes, often stirring to coat the orange slices evenly. Then remove from heat and cool for a couple of minutes.
Make the orange cake
Preheat the oven to 350F. Line a 9-inch springform pan with parchment paper and spray with nonstick cooking spray. Place the candied oranges over the parchment paper in the pan in a scaled spiral pattern.
In a food processor, add the zest of the remaining to the oranges. Then segment the oranges by removing any white pith, skin, and seeds, then cutting them into pieces. Add the segmented oranges to the food processor. Pulse into an applesauce-like consistency. Set aside.
In a big bowl or the bowl of a stand mixer, mix eggs and sugar until pale and fluffy. Slowly stream in the olive oil while still mixing. Add in the flour, salt, nutmeg, and baking powder and mix. Lastly, add in the yogurt, vanilla, orange mixture, and combine.
Pour the batter into the prepared baking pan. Bake for 50-55 minutes or until an inserted toothpick comes out clean.
Remove from the oven and let the cake cool in the pan for 30 minutes. Run a knife along the edges of the pan to release. Then release the spring on the pan. Slice and serve.