Preheat oven to 350°F. Grease a 13- x 18-inch rimmed baking sheet with nonstick spray (I recommend Baker's Joy). Set aside.
In a large mixing bowl, whisk together the butter, milk, pumpkin, and sugars. Whisk in the eggs and vanilla until combined. Whisk in the rest of the cake ingredients.
Pour batter into the pan. Bake on center rack 15-20 minutes, checking at 15 minutes to avoid over-baking. If it passes the toothpick test, it's done.
For the icing:
In a medium mixing bowl, combine the powdered sugar and cinnamon. Set aside.
As the cake is baking, melt the entire stick of butter in a pan over medium heat. Continue to heat until it turns a deep golden brown. It will foam and pop, so be careful.
Pour the brown butter into the bowl with the sugar and stir. Whisk in the vanilla and 4 tablespoons milk until the mixture becomes smooth and pour-able. Use more milk as needed, adding 1 tablespoon at a time.
Gently pour over the warm cake. Using a rubber spatula, quickly spread to cover the entire surface. The icing layer will be thin.
Serve immediately or store covered at room temperature up to 3 days.
Helpful tip: When pouring the icing on the cake, make sure to cover as much surface area as you can so it is easier to spread with the spatula.