Have ingredients at room temperature. Adjust rack to lower third of oven; preheat to 350°. Grease and flour a charlotte cake pan. Sift flour, baking powder, baking soda, cardamom, salt, ginger, and cinnamon onto waxed paper. In bowl of electric mixer using paddle attachment, beat butter on medium speed until creamy, 35-45 seconds. Add sugar; beat until fluffy, 5-6 minutes, occasionally scraping down sides of bowl. Add eggs, one at a time, beating well after each addition. Reduce speed to low; fold in ground walnuts and vanilla. Add dry ingredients in three additions, alternating with sour cream and ending with dry ingredients. Blend each addition just until incorporated. Spoon batter into pan. Bake until cake pulls away from sides of pan, 45-50 minutes. Cool upright in pan 10 minutes. Turn cake onto a rack over waxed paper; cool 1 hour. In a saucepan over medium heat, bring syrup to a boil; cook until reduced by half, 10-12 minutes. Brush syrup over cake, sprinkle with chopped walnuts and dust with confectioner's sugar.