Butter a 9-by-3-inch round cake pan. Make the caramel: Cook the sugar and corn syrup in a large skillet over medium-high heat, stirring occasionally, until the sugar is dissolved and the mixture is dark amber, 7 to 10 minutes.
Reduce the heat to medium. Carefully add the cream and 4 tablespoons butter (it will splatter) and cook, stirring, until combined, about 3 minutes.
Remove the skillet from the heat and let sit 1 minute. Pour 1 1/2 cups caramel into a glass measuring cup and set aside. Pour the remaining caramel into the prepared pan and set aside until set, about 30 minutes.
Cut 3 apples into quarters and cut out the cores with a paring knife. Arrange the pieces skin-side down in the pan (overlapping them slightly) to form a ring about 1/2 inch from the edge. Save any pieces that do not fit for later.
Halve the remaining whole apple crosswise. Scoop out the seeds with a melon baller or measuring spoon, leaving the stem on the top half. Chop the bottom and any of the leftover quartered apples into pieces; set aside.
Arrange the apple top, stem-side down, in the center of the pan. If the quartered apples shift, just push them back into place.
Preheat the oven to 350 degrees F. Make the batter: Whisk the flour, baking powder, baking soda, salt and cinnamon in a medium bowl. In a small bowl, whisk the sour cream, orange juice and vanilla.
Using a stand mixer with the paddle attachment, beat the butter and sugar on medium-high speed until light and fluffy, about 5 minutes.
Crack the eggs into a small bowl. Slip them into the mixer bowl, one at a time, and beat until the mixture is pale and creamy, about 5 more minutes. Scrape down the bowl and beater with a rubber spatula.
With the mixer on low speed, add half of the sour cream mixture, then half of the flour mixture. Repeat. Turn off the mixer and scrape down the bowl with a rubber spatula; finish combining the batter by hand.
Spread the batter over the apples in the pan. Top with the chopped apple. Bake on the middle oven rack until the cake is brown on top and springs back when pressed, 1 hour to 1 hour 20 minutes (don't worry if the top is dark). Cool in the pan on a rack.
Bring 1 inch of water to a simmer in a skillet wide enough to hold the cake pan. Carefully run a sharp knife around the inside of the pan several times, then rest the pan in the water to soften the caramel, about 8 minutes. Remove the pan from the water and dry.
Invert a plate on top of the cake, then flip over the cake and plate. Using pot holders or a towel, wriggle the pan off. Soften the reserved 1 1/2 cups caramel in the microwave, about 2 minutes. Drizzle the cake with some of the caramel. Slice and serve with the rest.