Preheat the oven to 350°F and grease an 8-inch round cake pan. Line the bottom with parchment paper.
In a large bowl, whisk together the flour, granulated sugar, cardamom, baking powder, baking soda, and salt.
In a separate bowl, beat the eggs with the vegetable oil and vanilla extract until smooth. Stir in the grated carrots.
Pour the wet mixture into the dry ingredients and stir just until combined. Spread the batter evenly in the prepared pan.
Bake for 28 to 32 minutes, or until a toothpick inserted in the center comes out clean. Let the cake cool completely in the pan before removing.
While the cake cools, melt the butter in a small skillet over medium heat. Cook until it turns golden brown and smells nutty, then pour into a bowl and let it cool to room temperature.
Beat the browned butter with the cream cheese until smooth. Add the powdered sugar and vanilla extract and mix until creamy.
Spread the frosting over the cooled cake and sprinkle the chopped toasted walnuts on top.