Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking dish with non-stick spray or butter.

In a mixing bowl, combine the yellow cake mix, 1 large egg, melted unsalted butter, pumpkin puree, and vanilla extract. Stir until the mixture is smooth and well-blended. Pour this batter into the prepared baking dish and press it evenly across the bottom.

In a separate bowl, beat the softened cream cheese until it's smooth and creamy. Add in the 2 large eggs, remaining pumpkin puree, and melted butter, mixing until fully combined. Incorporate the ground cinnamon, nutmeg, and cloves, then gradually add the powdered sugar, mixing until you achieve a silky filling. Pour this pumpkin mixture over the cake base.

Place the filled baking dish in the preheated oven and bake for 40-45 minutes. Once done, remove it from the oven and let it cool completely in the pan on a wire rack.

In a mixing bowl, use an electric mixer to beat together the softened cream cheese, unsalted butter, and brown sugar until light and fluffy. Gradually add the powdered sugar, mixing well until smooth. Stir in the vanilla extract.

Once the pumpkin spice gooey cake is completely cooled, spread the creamy frosting evenly over the top. Use a spatula to smooth out the frosting.