Cut each apple quarter lengthwise into 3 wedges. In a 10-inch cast-iron or nonstick skillet, over medium heat, melt 4 tablespoons of the butter. Stir in the brown sugar and cinnamon and cook 1 to 2 minutes, stirring, until the mixture is smooth. Remove from the heat and add the apples, gently stirring to coat. Return the pan to medium heat; cover and cook 5 minutes. Turn the apple slices over and cook 5 minutes more, uncovered; set aside. With a fork, arrange the apple slices in a spoke design in the sugar mixture, filling in the center with remaining slices.
Preheat oven to 350° F.
In a small bowl, combine the flour, baking soda, and salt; set aside. In a large saucepan, over medium heat, melt the remaining butter. Remove the pan from heat and whisk in the sugar, lemon juice, eggs, and vanilla extract. Whisk in the buttermilk and cornmeal. Add the flour mixture and stir gently with the whisk until the batter is smooth. Pour the batter over the apple slices. Bake about 35 minutes or until the cake springs back when gently pressed. Remove from oven and run a knife around the edge of the skillet to loosen the cake. Immediately cover the pan with a heatproof plate and carefully invert the two. Wait 1 minute before lifting off the skillet. Serve warm or at room temperature with whipped cream or a scoop of vanilla ice cream (if desired).