Preheat the oven to 350 degrees F. Grease a 18" X 13" jelly roll pan with nonstick cooking spray or butter and set aside.
Place the butter in a medium skillet. Melt the butter over medium heat, swirling it in the pan occasionally. The butter will foam and pop, so be careful. Continue to swirl the pan often. Remove pan from heat once the butter starts to brown and smells nutty. There will be small brown bits on the bottom. The butter should be an amber color. Pour butter into a small bowl and cool to room temperature. You can put it in the fridge to speed up the process.
While the butter is cooling, whisk together the flour, baking powder, baking soda, and pumpkin pie spice. Set aside.
In the bowl of a stand mixer, mix the brown butter and sugar together with the beater blade on medium speed. Mix until smooth. Add in the eggs, one at a time, and mix until combined. Beat in the pumpkin and vanilla extract. Mix until combined.
Turn the mixer to low and add in the dry ingredients. Mix until just combined, don't over mix. Stir the batter with a spatula to make sure all of the ingredients are combined.
Pour the batter into the prepared pan and use a spatula to make sure it is evenly spread out. Bake the cake for 18-24 minutes or until the cake springs back lightly when touched, is golden brown around the edges, and a toothpick inserted into the center comes out clean. Allow the cake to cool completely.
While the cake is cooling, make the brown butter cream cheese frosting. First, brown 1/2 cup of the butter. Only brown half of the butter for the frosting. Cut 1/2 cup of the butter into tablespoons and place it in a medium skillet. Melt the butter over medium heat, swirling it in the pan occasionally. The butter will foam and pop, so be careful. Continue to swirl the pan often. Remove pan from heat once the butter starts to brown and smells nutty. There will be small brown bits on the bottom. The butter should be an amber color. Pour butter into a small bowl and place in the freezer for about 20 minutes. You want it to solidify a little.
In the bowl of a stand mixer, beat together the cooled brown butter, regular butter, and cream cheese. Beat until smooth, about 3 minutes. Add the vanilla extract and pinch of salt and beat again.
Add powdered sugar one cup at a time, mixing well between each addition. Scrape down the sides of the bowl.
Frost the cooled pumpkin cake with the brown butter cream cheese frosting. Serve!
Note-cake will keep covered in the refrigerator for up to 3 days. Let it come to room temperature before serving. Although, I do like to eat it cold from the fridge:)