Preheat oven to 350 °F

Combine the cake mix, 1 egg and 1/2 cup (1 stick) butter and mix well with an electric mixer. Pat the mixture into the bottom of a lightly greased 13 by 9-inch baking pan.

In a large bowl, beat the cream cheese until smooth. Add the remaining 2 eggs, vanilla extract and remaining 1/2 cup (1 stick) butter and beat together until combined.

Slowly add the powdered sugar and mix well until smooth. Spread over cake batter and bake for 40 to 50 minutes. Center will still be a little gooey.

Let cool and set aside for churning portion of ice cream.

Set the 1/2 cup butter in a frying pan over low-medium heat. Stir occasionally as it melts and then it will begin to sputter and pop. Once the butter goes silent, and is foaming, stir more frequently to brown it evenly. When you see lots of brown bits at the bottom, immediately remove from heat and pour into a clean bowl, scraping the brown bits into the bowl. Let cool for a bit and the brown bits (milk solids) will sink to the bottom. Once they do, you can scoop off/pour off as much of the golden liquid from the brown butter bits as you can. Set aside the brown butter bits for step 2.

Add the brown butter bits, brown sugar, egg yolks, vanilla paste, heavy cream, milk, and salt to a pot. Whisk to combine.

Set the pot over medium-low heat and cook, stirring constantly until the temperature reaches 165 F. Immediately remove from heat.

Set a mesh sieve over a bowl and pour the custard through it to catch any bits of egg that might have cooked too quickly. Ideally pick a sieve that is not super fine so that it doesn't catch all of your brown butter bits!

Let the mix come to room temperature then cover with plastic wrap and set in the fridge for a few hours.

When ready to churn ice cream, give the custard a stir and then pour into prepared ice cream machine and churn according to manufacturer's instructions.

Moving back to your cooled gooey butter cake, crumble into large chunks.

When the ice cream is almost done churning, use a rubber spatula to gently fold in your cake chunks.

Store the ice cream in an airtight container and freeze. It’ll be ready to eat after freezing for about 4 hours.