Preheat the oven to 350°F and spritz a 15 x 10 x 1 ½-inch jelly roll pan with baking spray. Set aside.
In a medium size mixing bowl, sift together the flour, granulated sugar, brown sugar, pumpkin pie spice, baking soda and salt. (Tip: A wire whisk works perfectly)
In a separate 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 pan. Bounce on the counter to release trapped air.
Bake for 25 minutes or until a toothpick inserted into the center comes back clean. Cool completely in the pan on a cooling rack.
To make the frosting: In a heavy bottomed saucepan over low heat, melt the butter. Cook for 10 minutes or until the butter turns light golden brown 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 & 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.
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.