Preheat the oven to 350ºF. Nestle a piece of parchment paper into an 8- or 9-inch cast-iron pan and grease it with butter. In a bowl, whisk together the four, cardamom, baking powder, and salt. In a separate large bowl, beat the eggs until foamy. Whisk in the granulated sugar, cognac, and vanilla extract. Pour in the buttermilk and whisk to combine. Peel, halve, and core 2½ of the apples, then cut them into ½-inch-thick cubes. Reserve the remaining half apple, unpeeled, and cut it into ¼-inch slices, for the top.
Add half of the four mixture to the wet ingredients, stirring until just combined, then gently fold in half of the melted butter. Repeat with the remaining four mixture and melted butter.
Gently fold in the cubed apples, reserving the slices. Transfer the batter to the pan and arrange the apples slices in a circular pattern on top of the batter. Sprinkle with the turbinado sugar. Bake until the cake turns a deep golden brown and a cake tester inserted in the center comes out clean, 55 to 65 minutes. Transfer the skillet to a cooling rack and let it sit for 5 minutes.
The cake may be served warm or at room temperature, directly from the skillet; whichever you choose, make sure to dust it with confectioners’ sugar before serving.