How to Make It
Preheat oven to 350°F. Generously grease and flour a 14-cup Bundt pan. Heat cardamom pods in a small skillet over medium, stirring often, until lightly toasted and fragrant, about 2 minutes. When cool enough to handle, crack open outer shells, and transfer small brown seeds to a mortar and pestle. Discard shells. Crush seeds until finely ground; set aside.
Place granulated sugar, eggs, egg yolks, and salt in bowl of a heavy-duty stand mixer fitted with the whisk attachment. Beat on medium speed until mixture is light and fluffy, about 2 minutes. Beat in orange zest, vanilla, and ground cardamom, stopping to scrape down sides of bowl as needed. With mixer running on medium speed, gradually add olive oil in a slow, steady stream, slowing pour as necessary to ensure oil is fully incorporated.
Sift flour and baking powder over egg mixture. Using a rubber spatula, fold in flour mixture until almost completely incorporated; add apples, and fold just until combined. Spoon batter into prepared pan. Bake in preheated oven until golden brown and a long wooden pick inserted in center comes out clean, about 1 hour. Transfer cake in pan to a wire rack, and let cool 30 minutes. Invert cake onto wire rack, and cool completely, about 2 hours. Sprinkle with powdered sugar, and serve with whipped cream.
Cake may be wrapped in plastic wrap and stored at room temperature up to 2 days.