In a large microwave-safe bowl, combine the butter and sugars. Microwave for 30 seconds, then remove and stir with a whisk. Repeat process 2 more times. Though granules of sugar will remain, the goal is to get the mixture to a glossy finish.
Add the pumpkin puree and vanilla extract and mix well. Add the egg yolk. Mix well.
Add the flour, baking soda, baking powder, and salt and use a rubber spatula to fold everything together until no streaks of flour remain. Do not over mix.
Transfer the batter to the prepared pan and spread into a flat and even layer
In a clean bowl, combine the ingredients for the cheesecake swirl and mix until smooth and no lumps remain.
Drop the cream cheese mixture in big spoonfuls on top of the pumpkin batter. Use a butter knife to make big swirls. Do not do over swirl the bater–you want to big, distinct swirls.
Bake for 25 minutes or until the edges are set, and the center has puffed up, but is still slightly gooey.