Step 1: Prepare the Candied Oranges

1. Make the Syrup: Combine water and sugar in a medium saucepan. Bring to a boil over high heat, stirring until sugar dissolves.
2. Simmer Oranges: Lower heat to medium-low, add orange slices, and simmer for 15 minutes. Stir gently to coat slices evenly in syrup.
3. Cool Slices: Remove from heat and let the slices cool until they’re easy to handle, keeping them in the syrup. (Pro Tip: Use a slotted spoon or tongs to handle the orange slices for an easier transfer.)

Step 2: Prep the Cake Base

1. Preheat Oven: Set oven temperature to 350°F (175°C). Move an oven rack to the middle position.
2. Prepare Pan: Grease a 9-inch springform pan with cooking spray, line the bottom with parchment paper, and spray again. Arrange the cooled orange slices in an overlapping pattern at the base of the pan. Set aside.
3. Process Oranges: Zest two oranges, then remove their white pith. Cut into small pieces, discard seeds, and pulse in a food processor until you achieve a puree-like texture. (Common Mistake: Leaving too much pith can make the cake bitter; ensure it’s fully removed.)

Step 3: Mix the Batter

1. Whip the Eggs and Sugar: In a stand mixer or large bowl, whip eggs and granulated sugar until pale, fluffy, and doubled in volume (4-5 minutes). (Visual Cue: The mixture should form soft peaks when you lift the whisk.)
2. Stream in Oil: Gradually add olive oil while mixing on low speed.
3. Add Dry Ingredients: In a separate bowl, sift flour, salt, nutmeg (if using), and baking powder. Add dry mixture to wet ingredients in batches, mixing gently to combine. Avoid overmixing. (Pro Tip: Use the stir setting or mix by hand to prevent deflating the batter.)
4. Incorporate Yogurt and Flavorings: Fold in yogurt, vanilla extract, and the orange puree until just combined and the batter is smooth.

Step 4: Bake the Cake

1. Pour Batter: Gently pour the batter into the prepared pan, over the arranged orange slices. Smooth the top with a spatula.
2. Bake: Place in the oven and bake for 47-57 minutes. Check for doneness by inserting a toothpick into the center—it should come out clean or with a few crumbs. (Common Mistake: Overbaking can dry out the cake, so check frequently after 45 minutes.)
3. Cool: Let the cake cool in the pan for 30 minutes, then run a knife along the edges before releasing the springform. Cool completely before flipping onto a plate and peeling off the parchment paper.

Step 5: Serve or Store

1. Brush Syrup (Optional): For a glossy finish, brush the leftover orange syrup over the cake’s surface.
2. Slice and Serve: Use a serrated knife to cut clean slices, and serve at room temperature. (Serving Suggestion: Pair with whipped cream, a scoop of gelato, or a drizzle of dark chocolate sauce.)