Preheat oven to 350 degrees F. Spray a half sheet sheet pan with nonstick cooking spray and set aside.
In a large mixing bowl whisk together the flour, baking powder, baking soda, salt, and pumpkin pie spice. Set aside
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter and sugar on medium speed until combined and creamy, about 1 minute. Add the eggs and vanilla and beat on low until combined, about 30 seconds. Beat in the pumpkin on low until just combined. Add the flour and pulse the mixer until half of the flour is mixed in. Increase the speed to low and mix until just combined.
Pour the batter into your prepared sheet pan and spread evenly with an offset spatula. Bake for 25-30 minutes, or until a toothpick inserted comes out with just a few moist crumbs. Place the sheet pan on a wire rack to cool completely, about 2 hours.
For the Icing:
In a medium saucepan over medium-high heat, melt the butter and continue to cook, swirling the pan constantly until the butter is dark golden brown (not burnt) and has a nutty aroma. Remove the saucepan from the heat and whisk in the brown sugar, salt, 2 tablespoons of the milk, maple syrup, and vanilla.
Transfer the mixture to the bowl of a stand mixer fitted with a paddle attachment and beat on medium speed for 30 seconds. Let the mixture rest for 3 minutes, then mix again on medium speed for 30 seconds. Repeat for one more round of resting and mixing. Add the sifted powdered sugar and cinnamon, pulse until almost entirely combined, then increase to medium and beat until creamy. Add milk, 1/2 tablespoon at a time, until the desired consistency is reached.
Using an offset spatula, spread the icing over the cake. Let set up for 30 minutes before slicing into 24 squares.