Preheat the oven to 350°F / 175°C. Grease an 8-by-8-inch square baking pan and line it with parchment paper, leaving an overhang on two sides for easy removal.
Make the streusel topping. In a medium bowl, mix the flour, brown sugar, granulated sugar, cinnamon, pumpkin pie spice, and salt. Add the cold butter cubes. Use a fork, pastry cutter, or fingertips to work the butter into the dry ingredients until the mixture forms coarse crumbs with some larger clumps. Set aside.
In a medium bowl, whisk together the flour, pumpkin pie spice, baking powder, baking soda, and salt for the cake.
In a large bowl, whisk the melted butter, brown sugar, granulated sugar, pumpkin puree, sour cream, eggs, and vanilla extract until smooth.
Add the dry ingredients to the pumpkin mixture. Fold with a spatula just until combined and no dry streaks remain. Do not overmix.
Spread the batter evenly in the prepared pan.
Sprinkle the streusel topping evenly over the batter, leaving some larger clumps intact.
Bake for 35 to 40 minutes, until the topping is golden and a toothpick inserted into the center of the cake comes out clean or with a few moist crumbs.
Cool the cake in the pan for at least 20 minutes before slicing into 9 squares and serving.