Beat 3 cups confectioners' sugar, cream cheese, lemon juice, and 1 teaspoon vanilla extract together in a large bowl using an electric mixer until frosting is smooth. Cover bowl with plastic wrap and refrigerate while preparing cake.
Preheat oven to 350 degrees F (175 degrees C). Butter a 9x13-inch baking pan.
Beat brown sugar and butter together in a large bowl using an electric mixer until smooth; beat in eggs, 1 at a time, until smooth. Stir 2 tablespoons ginger, 1 teaspoon vanilla extract, and salt into brown sugar-butter mixture until incorporated.
Beat flour and baking powder into brown sugar-butter mixture until batter is smooth; fold in 3/4 cup dried cranberries and white chocolate chips. Pour batter into the prepared baking pan, spreading evenly.
Bake in the preheated oven until cake is lightly browned and a toothpick inserted in the center comes out clean, about 30 minutes. Cool cake completely, 4 hours to overnight. Remove frosting from refrigerator 1 hour before frosting the cake.
Spread frosting evenly over cake; sprinkle with 1/4 cup dried cranberries and 2 tablespoons ginger.
Whisk 1/2 cup confectioners' sugar and cream together in a bowl until smooth; drizzle over frosted cake. Cover cake with plastic wrap and refrigerate until chilled, about 2 hours.
Cut cake in half lengthwise and each half into 8 equal rectangles. Cut each rectangle in half diagonally, creating 16 triangle-shaped bars.