Preheat oven to 350°F. Grease a 10x15x1 baking pan, set aside.
In a large bowl, combine flour, baking powder, baking soda, spices, and salt.
In a separate bowl whisk melted butter, granulated sugar, and brown sugar. Add eggs, pumpkin, and vanilla and whisk until well combined.
Pour the wet ingredient into the dry ingredients and whisk until just combined. Don't whisk too vigorously or for too long (you don't want to add too much air into the batter).
Pour the batter into the prepared pan and smooth it into an even layer. Bake for 18-22 minutes, until a toothpick inserted into the center comes out clean. Place the pan on a wire rack to cool completely. Note: The cake will reach the top of the pan while baking.
Brown butter frosting:
Melt the butter in a saucepan over medium heat (see tips for browning butter in the notes section). Swirl the pan occasionally to be sure the butter is cooking evenly. As the butter melts, it will begin to foam. Watch the butter closely, continuing to swirl the pan occasionally. It will eventually turn golden brown with brown bits on the bottom of the pan. Immediately pour the butter (and brown bits) into a large mixing bowl. Allow to cool to room temperature.
Using a stand mixer (fitted with the paddle attachment) or a hand mixer, beat the cooled butter with 2 cups of powdered sugar. Add the half and half, vanilla, and additional cup of powdered sugar and beat until well combined. Add more half and half if you need to thin the frosting or more powdered sugar to thicken it. Once you get it to your desired consistency beat it on high speed for about 20-30 seconds. Spread the frosting on the cooled cake.
Notes
Tip for browning butter:
Use a light-colored saucepan so that you can watch the color of the butter closely.
Swirl the pan occasionally to be sure the butter is cooking evenly.
The butter will take a while to turn brown, but once it is brown it can burn quickly, so make sure you're watching it closely.
As soon as the butter has browned move it out of the hot pan and into a bowl to cool.