Heat oven to 350° F (176° C). Prepare a deep 8-inch round cake pan by coating the inside with vegetable shortening or butter. Line the bottom with parchment paper, grease the parchment paper, and then coat the inside of the pan with flour. (Here's how to prepare a cake pan so the cake won't stick.)
Add the brown sugar, both flours, and melted butter to a bowl and stir by hand or with an electric mixer fitted with the paddle attachment on medium-low speed, until the mixture resembles course crumbs.
Remove 1 cup of the crumb mixture and press it into the bottom of the cake pan in an even layer.
Add the remaining ingredients to the mixing bowl with the rest of the crumb mixture and mix on low speed just until the batter is smooth. Scoop the batter into the cake pan, spreading evenly it over the crumb layer.
Bake the cake in the center of the oven for 60-70 minutes, until a toothpick inserted into the center of the cake comes out without any raw batter on it. Let the cake cool in the pan for about 5 minutes then gently turn it out onto a wire rack to allow it to cool completely before frosting.
Add all the buttercream ingredients to a bowl and beat with an electric mixer until completely smooth and fluffy - about 5 minutes. Spread the buttercream over the top of a completely cooled pistachio cardamom cake.
