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Preheat the oven to 350 degrees F. Grease a 9-inch springform pan and set aside.
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For the streusel topping/filling:
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1. In a small bowl, whisk together brown sugar, granulated sugar, flour, oats, and cinnamon. Cut in the cold butter with a pastry blender until crumbly then set aside.
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For the cake:
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2. In a large bowl, cream together butter and granulated sugar until fluffy. Add the eggs one at a time, mixing until incorporated. Stir in the vanilla extract. Set aside.
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3. In another bowl, add the dry ingredients: the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves. Whisk to combine.
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4. Gradually add the dry ingredients to the creamed mixture, stirring until just incorporated. Scoop the squash puree into the mixing bowl, and mix until that batter just comes together.
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5. Spoon half of the cake batter into the prepared springform pan, smoothing the batter to level.
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6. Spread the applesauce over the batter in the pan.
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7. Sprinkle half of the streusel over the apple sauce.
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8. Spoon the remaining batter evenly over the streusel, and smooth to the edges of the pan.
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9. Sprinkle the remaining streusel on top of the batter.
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10. Bake at 350 degrees F for 50- to 55 minutes, or until a toothpick inserted into the center of the cake comes out clean.
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11. Cool cake in the pan for 10 minutes before removing the side of the pan.
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12. Allow cake to cool completely on the base of the springform pan, set on a wire rack.
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13. Once completely cool, transfer coffee cake to a serving plate, and prepare the glaze.
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For the glaze:
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1. In a small bowl, use a fork to stir together the powdered sugar, hot water, and vanilla extract, until smooth.
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2. Use the same fork to drizzle the glaze over the cooled coffee cake.
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Serve immediately. Leftover coffee cake will keep for 3 to 4 days if tightly wrapped and stored at room temperature.