Preheat your oven to 350 degrees F. Grease a 9×13 inch cake pan with nonstick spray or butter.
Make the streusel: In a large bowl, whisk together 3 cups flour, 1 cup brown sugar, 1/2 cup granulated sugar, 1 teaspoon kosher salt, 1 tablespoon cinnamon, and 1 tablespoon pumpkin pie spice. Melt 1 cup butter in a medium bowl in the microwave. Pour the melted butter into the large bowl with the flour mixture and stir to combine. The mixture should resemble wet sand with chunks. Some loose flour at the bottom is fine. Set aside; refrigerate if space allows.
For the cake: In a large bowl or stand mixer, whisk together 2 1/4 cups flour, 1 1/2 cups granulated sugar, 1 tablespoon baking powder, 3/4 teaspoon baking soda, 3/4 teaspoon kosher salt, 1 tablespoon pumpkin pie spice, and 1 teaspoon cinnamon. Chop softened butter into chunks, making sure it is soft but not creamy. Add chopped butter to the flour mixture.
Open the can of pumpkin puree and scoop out about half of it; add it to the flour and butter mixture. Use the paddle attachment or beaters to blend until the mixture comes together in a ball and the butter is fully incorporated. Scrape the bowl sides and bottom as needed. Add a bit more pumpkin puree if necessary to help combine.
In the medium bowl used for melting butter, add the remaining pumpkin puree. Whisk in 4 eggs, 1/4 cup vegetable oil, and 1 tablespoon vanilla extract until mixture is smooth and well blended.
Add the egg mixture to the flour mixture in 3 additions, beating about 20 seconds after each addition. After all is combined, beat for an additional minute until the batter is light and fluffy.
Spread 2 cups of batter evenly into the prepared cake pan. Sprinkle 1 cup of streusel evenly over the batter, breaking up large pieces by hand. Spread another 2 cups of batter on top and smooth as best as possible. Sprinkle another 1 cup streusel over this layer. Add the remaining batter and spread evenly. Finally, spread another 2 cups streusel over the top of the cake. Do not add all remaining streusel; save leftovers for later and refrigerate if possible.
Bake the cake at 350°F for 35 minutes.
After 35 minutes, quickly remove the cake from the oven and close the oven door. Quickly sprinkle the remaining streusel over the cake, focusing on spots where the batter has puffed up. Return the cake to the oven promptly and keep the oven door closed as much as possible.
Bake for an additional 10-15 minutes. Total baking time should be about 45-50 minutes depending on your oven and pan type. Check doneness by inserting a toothpick in the center — it should come out clean without wet batter. The cake should be stable and not wobble when shaken gently.
Allow the cake to cool on a wire rack for 15-20 minutes before slicing.
Make the icing: In a medium bowl, beat 1/4 cup very soft butter until smooth. Add 1 teaspoon vanilla, 1/2 teaspoon kosher salt (reduce if using table salt), 3 tablespoons milk, and 2 cups powdered sugar. Beat until smooth and creamy. Adjust consistency by adding milk 1-2 teaspoons at a time if icing is too thick.
Drizzle the icing over the cake. You may cover the entire cake or drizzle icing on individual slices. For best results, drizzle half the icing over the whole cake, reserving the rest for serving portions.
Serve the cake warm. If serving after cooling completely, heat individual slices for 10-