Preheat oven to 350 degrees F. Prepare a bundt pan by heavily greasing with butter and dusting with a thin layer of flour.
To make the crumble, mix brown sugar and flour in small bowl. Use fork or pastry blender to cut in cold butter to make course crumbs. Mix in nuts, cinnamon, and pumpkin pie spice. Set aside.
To make the cake, mix dry ingredients (flour, baking powder, cinnamon, baking soda, pumpkin pie spice, and salt) in small bowl. Set aside.
In stand mixture, cream butter and sugar on high speed until very well combined, light, and fluffy. Beat on high for five minutes, using a scraper to loosen sides and bottom half way through. Once thoroughly combined, scrape sides again. Slowly mix in eggs, one at a time. Once eggs have started to combine, turn the speed up to medium and beat for about 30 seconds. Once again, scrape sides and bottom of bowl. Add pureed squash and vanilla. Stir on low and then once combined, increase speed to medium and mix for about 30 seconds until well combined. Scrape bowl.
With speed set to low and the mixer running, scoop dry ingredients into bowl one spoonful at a time. Once all the dry ingredients have been added and are somewhat combined, increase speed to medium to thoroughly mix, about 30 seconds. Batter will be very thick and sticky.
Scoop half the batter into the prepared pan and spread the best you can. Pour the applesauce over the batter and spread to create an even layer. Sprinkle crumb mixture evenly over applesauce. Finally, scoop remaining cake batter on top of the crumb mixture and do your best to spread it evenly. The cake mix may not entirely cover the crumb mixture.
Bake in preheated oven for 50-55 minutes or until a knife inserted comes out clean. Remove from oven and allow to cool in pan slightly, about 5 minutes, before inverting onto plate.
Whisk together glaze ingredients and drizzle over inverted cake. Serve warm or at room temperature. Store in airtight container.