Preheat the oven to 350°F (175°C). Line an 8x8-inch baking pan with parchment paper, leaving some overhang on the sides for easy removal.
Melt the butter in a saucepan over medium heat. Continue cooking, swirling the pan occasionally, until the milk solids turn golden brown and the butter smells nutty. Remove from the heat and let it cool slightly. Reserve ¼ cup of the browned butter for the frosting.
In a medium bowl, whisk together the flour, pumpkin pie spice, baking soda, and salt until evenly combined.
In a large bowl, combine 1 cup of the browned butter, granulated sugar, and brown sugar. Whisk until well combined and smooth.
Add the pumpkin puree, eggs, and vanilla. Whisk until the mixture is smooth and evenly combined.
Add the dry ingredients to the pumpkin mixture and fold together just until no streaks of flour remain. Don't overmix.
Pour the batter into the prepared 8x8-inch pan and use a spatula to spread it evenly into the corners.
Bake for 25–30 minutes, until the edges are set and lightly golden and the center no longer looks wet. A toothpick inserted into the center should come out with a few moist crumbs, not wet batter. Don't wait for it to come out completely clean.
Let the blondies cool completely in the pan on a wire rack before frosting. Once cool, use the parchment overhang to lift them from the pan.