Preheat the oven to 350°F and spritz a 15x10x1 1/2-inch jelly roll pan with baking spray. Set aside.
In a medium size mixing bowl use a whisk to sift together the flour, granulated sugar, brown sugar, pumpkin pie spice, baking soda and salt. (Tip: A wire whisk works perfectly)
In a separate large mixing bowl, whisk together the pumpkin puree, eggs, oil and vanilla.
Using an electric mixer on low gradually add the dry ingredients to the wet ingredients. Mix until fully moistened and the batter is creamy.
Pour batter evenly into the prepared pan. Bounce on the counter to release trapped air.
Bake for 25 minutes or until a toothpick or cake tester inserted into the center comes back clean. Cool to room temperature in the pan on a wire rack.
Brown Butter Frosting: In a heavy bottomed saucepan over low heat, melt the butter. Cook for 10 minutes or until the butter turns light golden brown (amber) in color, stirring occasionally. The butter will bubble and froth.
Once browned, remove from the heat. Set aside to cool slightly.
Add the powdered sugar, half and half and vanilla to a medium size mixing bowl. Using an electric mixer on low speed gradually add the browned butter. After all is added, increase the speed of the mixer and beat until fluffy and smooth. Add additional half and half in 1 Tbsp increments if needed, until it reaches spreading consistency.
Frost the cooled cake. Sprinkle with 1 cup toasted pecans or walnuts if desired.
Store chilled.
Toasted Pecans: To toast the nuts, place 1 cup chopped nuts in a single layer on a baking sheet. Toast in a 350°F oven for 6-8 minutes. Cool completely before sprinkling on top of the cake.