Make the apple pie filling: In a medium pan, melt butter over medium heat. Add diced apples, brown sugar, cinnamon, nutmeg, and lemon juice. Cook until apples are soft (6-8 minutes). Stir in cornstarch-water mixture and let it simmer until thickened. Remove from heat and allow to cool.
Make the cake: Preheat the oven to 350°F (175°C). Line a 10x15-inch jelly roll pan with parchment paper and spray with nonstick spray. In a bowl, combine flour, baking powder, salt, cinnamon, and nutmeg. In another bowl, beat eggs, granulated sugar, and brown sugar until light and fluffy (3-4 minutes). Add vanilla and milk, then mix. Gradually add the dry ingredients until just combined. Pour the batter into the pan and spread evenly. Bake for 10-12 minutes, until the top springs back when touched.
Roll the cake: Lay a clean kitchen towel on your counter and dust generously with powdered sugar. Once the cake is done, turn it out onto the towel. Peel off the parchment paper and roll the cake while it’s still warm, starting from the short side. Let it cool completely while rolled up.
Fill and roll again: Once the cake has cooled, unroll it gently. Spread the apple pie filling evenly over the cake. Carefully roll it back up (without the towel). Place the roll seam-side down on a serving tray.
Add caramel: Warm the caramel sauce if needed and drizzle it generously over the cake. Optionally, sprinkle a little sea salt for a sweet-salty twist.
Chill and serve: Refrigerate the cake for at least 30 minutes to firm up. When ready to serve, slice with a sharp knife and top with whipped cream if desired.
