In a medium-sized saucepan, combine water, granulated sugar, and corn syrup. Use at least a 2-quart pan to prevent overflow.

Over medium heat, bring the sugar mixture to a boil. Do not stir during this process to avoid crystallization.

Let the mixture boil until it turns golden brown, around 5-7 minutes. Watch carefully to avoid burning.

Warm the cream in the microwave for 1 minute.

Once the caramel achieves a golden brown color, remove from heat and carefully add the warm cream. Stir until combined.

Transfer the caramel mixture to a separate bowl, then add vanilla, salt, and butter. Let it sit until the butter melts, and then stir until well incorporated.

Let it sit until room temperature before adding to the gooey top. Make sure the caramel is cool and not warm.

Preheat your oven to 350°F (175°C). Grease a 9-inch cake pan or springform pan with cooking spray, line with parchment paper, and spray again, ensuring the sides are covered. Set aside.

In a medium-sized bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment, cream the butter, oil, and sugar on high until fluffy and smooth, about 4 minutes. Scrape the bottom and sides of the bowl.

Add the whole eggs and egg yolk one at a time, beating on medium speed after each addition for about 30 seconds. Scrape the bowl after each addition. Once all the eggs are added, beat on medium-low for 30 more seconds.

Add the butter extract and mix on medium-low speed for 30 seconds. If the mixture looks clumpy, it will smooth out once you add the flour mixture.

In a small bowl, combine the sour cream and buttermilk until well mixed.

Turn the mixer to low and add half of the flour mixture and half of the buttermilk mixture to the butter mixture. Mix on low until almost incorporated. Add the rest of the flour and buttermilk mixtures and mix just until a few streaks of flour remain. Gently mix the bottom and sides of the bowl with a rubber spatula.

Distribute the cake batter into the prepared springform pan. Set aside while you make the gooey butter topping.

In a medium-sized bowl, mix the cream cheese with a hand mixer on medium-high speed until smooth and creamy, about 30-45 seconds.

Add the egg and mix on high for 30 seconds. Scrape the sides and bottom of the bowl with a rubber spatula.

Add the sifted powdered sugar, 1 cup at a time, mixing for 30 seconds after each cup has been added. This should take a little over 1 minute.

Add the caramel sauce and mix on low just until incorporated.

Pour the cream cheese mixture onto the top of the cake batter and use a small offset spatula to spread it into an even layer.

Bake for 45-48 minutes. The middle of the cake will still be jiggly, but the cake is done when the edges are golden and the middle has cracked.

When the cake is done, let it cool for 10-15 minutes, then place it in the fridge to chill for at least 2 hours to allow the cream cheese layer to firm up and set.

Dust with powdered sugar before serving.