Preheat the oven to 350℉. Spray an 8x8 inch pan with cooking spray and line it with parchment paper. Add the butter into a pan and allow it to melt over medium heat. Continue to stir and cook the melted butter until it develops a golden brown color, about 5-8 minutes.
In the meantime, add the flour, pumpkin pie spice, kosher salt, and baking powder into a bowl. Mix with a whisk until the ingredients are evenly distributed. Add the browned butter into a large bowl along with the dark brown sugar and granulated sugar. Whisk until the ingredients are well combined. Then mix in the egg, vanilla extract, and pumpkin puree.
Pour the dry ingredients into the bowl of wet ingredients. Mix until the batter is evenly combined. Pour half of the batter into the prepared baking pan and spread it out into an even layer. Add the Oreos onto the batter in a single layer, slightly push them into the batter. Spread the remaining batter on top. Sprinkle the crumble mixture over top until fully covered.
Bake it in the oven for 25-30 minutes. When a toothpick is inserted into the center, it should come out clean. Allow it to cool completely before cutting it into squares and serving.