Step 1: Preheat your oven to 350°F (175°C). Grease and flour a 9-inch cast-iron skillet or a round baking pan.
Step 2: In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cardamom, and salt. Set aside.
Step 3: In a large bowl, using an electric mixer, cream together the softened butter, granulated sugar, and light brown sugar until light and fluffy, about 2-3 minutes.
Step 4: Beat in the eggs one at a time, ensuring each is fully incorporated before adding the next. Stir in the vanilla extract and cognac.
Step 5: Gradually add the dry ingredients to the wet ingredients in three additions, alternating with the buttermilk in two additions, beginning and ending with the dry ingredients. Mix on low speed until just combined, being careful not to overmix.
Step 6: Gently fold in the diced apples until evenly distributed throughout the batter.
Step 7: Pour the batter into your prepared skillet or baking pan, spreading it evenly. Sprinkle the turbinado sugar over the top of the cake batter.
Step 8: Bake for 45-55 minutes, or until a wooden skewer or toothpick inserted into the center comes out clean. The top should be golden brown and the edges slightly pulling away from the pan.
Step 9: Let the cake cool in the pan on a wire rack for at least 15-20 minutes before serving warm, or allow it to cool completely.