Preheat the oven. Set the oven to 350°F and line an 8×8-inch baking pan with parchment paper.
Prepare the crumble. In a bowl, combine the light brown sugar, flour, and pumpkin pie spice. Add the cold salted butter and use a pastry cutter or your hands to work it into the dry ingredients until crumbles form. Place the mixture in the refrigerator while preparing the cake batter.
Mix the wet ingredients. In a large bowl, combine the melted salted butter, pumpkin puree, granulated sugar, eggs, and vanilla extract.
Mix the dry ingredients. In a separate bowl, combine the flour, baking powder, baking soda, salt, and pumpkin pie spice. Add the dry ingredients to the wet mixture and mix until combined.
Layer the cake. Spread half of the pumpkin batter into the prepared baking pan. Sprinkle half of the chilled brown sugar crumble evenly over the batter.
Add the remaining layers. Spread the remaining pumpkin batter over the crumble layer, then distribute the rest of the crumble across the top.
Bake. Place the pan in the oven and bake for 35–40 minutes. Check the center with a toothpick for wet batter, or gently press the cake to see if the center springs back.
Cool and serve. Remove the coffee cake from the oven and allow it to cool slightly before slicing and serving.