Preheat oven to 350°F. Line an 8″x8″ dish with parchment paper and set aside.
In a microwave-safe bowl, combine butter and white chocolate chips and microwave for 30 seconds. Stir the mixture until completely melted (if not melted, give it another 10-15 seconds).
Add in egg, sweetened condensed milk, vanilla extract, and salt, stirring to combine. Fold in flour, stirring until just combined. Separate out 1 1/2 cups of the mixture and stir in the pumpkin puree, cinnamon, and nutmeg. Pour the white chocolate mixture into the prepared pan, then dollop with the pumpkin mixture. Swirl the pumpkin into the white chocolate using a knife.
Bake for 30 minutes. It should be slightly puffed up in the middle (this will sink back down when you remove it from the heat). Remove from oven, allow to cool in pan for 10 minutes, then transfer the brownies out of the pan and onto a cooling rack. Allow to cool completely before serving.