In a medium bowl, whisk together the flour, walnuts, brown sugar, and pumpkin pie spice. Stir in the melted butter. Cover and refrigerate the streusel for at least 15 minutes and up to 1 day.

Preheat the oven to 350°F. Spray a 13-by-9-inch baking pan with baking spray. Line the pan with parchment paper, leaving a 1-inch overhang. Spray the parchment.

In a large bowl, whisk together the pumpkin puree, brown sugar, vegetable oil, sour cream, granulated sugar, vanilla, and eggs until well combined. Add the flour, pumpkin pie spice, baking powder, baking soda, and salt, stirring until just combined.

Pour the batter into the pan, smoothing the top with an offset spatula. Remove the cold streusel from the refrigerator, and use your fingers to break it into dime-size pieces. Sprinkle the streusel evenly over the batter.

Bake the cake until a toothpick inserted in the center comes out with a few moist crumbs, 40 to 45 minutes. Let cool completely in the pan.

In a medium heatproof bowl, microwave the cream cheese until very soft, 10 to 15 seconds. Whisk in the powdered sugar, vanilla, and 1 tablespoon water.

Using the parchment overhang, lift the coffee cake from the pan. Drizzle the cream cheese drizzle over the cake. Let set for 15 minutes before slicing.