Preheat your oven to 350°F.
Mix together all ingredients for the cake, and pour the batter into the springform mold (A springform pan is a two piece pan composed of an expandable ring with a release latch for easy removal with an interconnected and removable bottom).
Peel, core, and quarter the apples. Then cut them into wedges. Place the apple wedges in the pan, like when doing a tart or a pie, starting on the outer edge of the pan and moving to the center.
Heat the pan over medium heat and melt the butter. Sprinkle the sugar evenly over the melted butter and cook, stirring occasionally until the mixture is a very pale amber. Off the heat, add the lime zest and vanilla and whisk to lower the temperature, then add the egg and whisk to avoid getting scramble eggs. You will get a shiny sauce fill with richness and flavor.
Pour this butter sauce over the cake batter and apples cover well. then cook in the oven for 20 minutes.
Serve warm or at room temperature.