Preheat the oven to 350 degrees. Grease and flour an 8 inch square cake pan. Set aside.

Place the butter in a medium saucepan or skillet and heat over medium low heat, swirling frequently until the butter has a nutty aroma and is golden brown. Transfer brown butter to a bowl to cool.

Combine flour, baking powder, baking soda, salt, cinnamon, nutmeg, and cardamom in a medium bowl. Whisk together and set aside.

Combine eggs, brown sugar and granulated sugar in a large bowl and whisk until fully combined and no lumps remain. Slowly whisk in oil followed by cooled brown butter and vanilla extract until emulsified.

Add dry ingredients to wet ingredients and stir until just a few streaks of flour remain. Fold in carrots until batter is combined.

Transfer cake batter to prepared pan. Bake for 30-40 minutes or until a tester comes out clean or with a few moist crumbs. Set cake aside to cool.

To make frosting: Combine butter and cream cheese in a stand mixer bowl. Beat on medium until fully combined, scraping bowl as needed.

Reduce mixer speed to low and add powdered sugar, vanilla, and a pinch of salt. Beat until frosting is smooth, about 1-2 minutes, scraping bowl as necessary.

Once cake is completely cool, spread frosting evenly on top.

Serve or cover and refrigerate until ready to serve.