For the spiced carrot cake: Preheat the oven to 350 degrees F. Butter a 9-by-9-by-3-inch square baking pan. Line the pan with parchment paper and butter and flour the paper.
Brown the butter in a small stainless-steel saucepan over medium-high heat. It should melt, foam and then the foam should subside. When you smell a nutty aroma, remove the butter from the heat. Remember that it will continue to brown as it cools.
Combine the flour, baking powder, baking soda, cardamom, cinnamon, cloves and salt in a medium bowl. Whisk together the olive oil, vegetable oil, browned butter and vanilla in a medium bowl. Mix the eggs, granulated sugar and brown sugar in an electric mixer until fluffy, about 4 minutes.
Drizzle the oil mixture into the mixer and whisk until light and fluffy, 2 to 3 minutes. Transfer to a large mixing bowl and fold the dry ingredients into the mixture with a rubber spatula until combined, then stir in the carrots. Pour the batter into the prepared pan and bake until a toothpick inserted into the center comes out clean, about 1 hour and 15 minutes.
For the frosting: Dissolve the espresso powder in 2 tablespoons of hot water. Beat the cream cheese and butter in a large bowl with an electric mixer on medium speed until fluffy, about 3 minutes. Add the confectioners' sugar, espresso and cardamom and beat until light and fluffy.
To assemble: Remove the cake from the pan and let cool on a wire rack. Cut into 9 squares, then split each square in half horizontally. Spread some of the icing between two cake layers, then generously spread icing over the top and garnish with the candied walnuts.